Combinatorics, Graph Theory and Computing: SEICCGTC 2022, Boca Raton, USA, March 7–11 (Springer Proceedings in Mathematics & Statistics, 462) [2024 ed.] 3031621654, 9783031621659

Table of contents :
Preface
Volume Overview
Contents
Additive Combinatorics in Groups and Geometric Combinatorics on Spheres
1 Introduction
2 The Independence Number of a Subset of an Abelian Group
3 Spherical Designs
4 Spherical Few-Distance Sets
5 The Spanning Number of a Subset of an Abelian Group
References
A Walk Through Some Newer Parts of Additive Combinatorics
1 Introduction
2 Minimum Sumset Size
3 Perfect Bases and Spanning Sets
4 (k,l)-Sumfree Sets
5 Sumset Size of Maximum-Size Nonbases
References
Passing Drops and Descents
1 Introduction
2 Generalization of Eulerian Numbers
3 Describing Passing Patterns with Cards
4 Describing Passing Patterns with Rook Placements
4.1 Avoiding Throws of 0
References
Group Divisible Designs with Three Groups and Block Size 4
1 Introduction
2 The Necessary Conditions for GDD (1, n, n + 2, 4; λ1,λ2)
3 General Construction of GDD(1, n ,n + 2,4;λ1, λ2)
3.1 GDD(1, n ,n + 2,4;λ1, λ2) When λ1 ≤3
3.2 GDD(1, n, n+2, 4; λ1, λ2) When λ1λ28mu(mod6mu6)
3.3 GDD(1,n,n+2,4;λ1,λ2) with the Cases on the Off-Diagonal Entries of Table 1
4 GDD(3, n, n + 1, 4;λ1,λ2)
4.1 Necessary Conditions for GDD(3, n, n + 1, 4;λ1,λ2)
5 GDD(3, n, n + 1, 4;λ1,λ2) with Small Parameters
5.1 GDD(3, 3, 4, 4;λ1,λ2)
5.2 GDD(3, 4, 5, 4;λ1,λ2)
5.3 GDD(3, 6, 7, 4;λ1,λ2)
6 Summary
Appendix: Non-Existence Result of GDD(1, n, n + t, 4; λ1, λ2) with Configuration Restrictions
Equal Number of Blocks of Type (1, 1, 2) and (2, 2)
Equal Number of Blocks of Type (1, 1, 2) and (0, 4)
Equal Number of Blocks of Type (0, 4), (1, 1, 2) and (2, 2)
References
A New Absolute Irreducibility Criterion for Multivariate Polynomials over Finite Fields
1 Introduction
2 Background
3 Results
4 Algorithm
References
On Combinatorial Interpretations of Some Elements of the Riordan Group
1 Introduction
1.1 The Riordan Group
1.2 Generalized Ordered Trees
2 Riordan Arrays Obtained from the Bell Subgroup
2.1 Generalized Dyck Paths and k-Trees
2.2 Riordan Arrays, k-Trees, and k-Dyck Paths
3 Inverses of Elements of the Bell Subgroup
References
Production Matrices of Double Riordan Arrays
1 Introduction
2 Double Riordan: A and Z Sequences
3 The Production Matrix
4 Combinatorial Example
References
The Existence of a Knight's Tour on the Surface of Rectangular Boxes
1 Introduction
2 Preliminaries
2.1 Definitions
2.2 Theoretical Preliminaries
2.3 Knight's Tours on Boards with Odd Dimensions
3 Proof of Main Theorem
3.1 When m≥5 and at Least One Element of {m,n,r} Is Even
3.2 When m≥5 and Three Side Lengths Are Odd
3.3 When m=4
3.4 When m=3
3.5 When m=2
3.6 When m=1
4 Open Problems
4.1 Generalized Knight
4.2 Generalized Board
Appendix
References
A New Upper Bound for the Site Percolation Threshold of the Square Lattice
1 Introduction
2 Background
2.1 Matching Lattices and Line Graphs
2.2 Substitution Method
3 Application of the Method
3.1 Reference Lattice
3.2 Two Stage Process
3.3 Boundary Vertex and Partition Issues
4 Summary and Continuing Research
References
Prime, Composite and Fundamental Kirchhoff Graphs
1 Introduction
2 Finding Kirchhoff Graphs Using Uniformity
2.1 Structure of the Algorithm
2.2 Kirchhoff Graph Examples Found by the Algorithm
3 Tiling of Kirchhoff Graphs
4 Fundamental Graphs, Tiling Structure
5 Open Questions
References
On the Minimum Locating Number of Graphs with a Given Order
1 Introduction
2 Simple Results
3 Minimum Locating Number
References
j-Multiple, k-Component Order Neighbor Connectivity
1 Introduction
1.1 Applications
1.2 Terminology
2 General Results
3 Results for Specific Graph Types
3.1 Complete Graphs and Complete Bipartite Graphs
3.2 Paths, Cycles, and Wheels
3.3 Complete Grid Graphs
References
Graphic Approximation of Integer Sequences
1 Introduction
2 Definitions and Needed Results
3 Algorithm 1—Minimizing the Discrepancy
3.1 Finding a Minimal Discrepancy Graphic Sequence
3.2 Creating a Fast Algorithm
4 Algorithm 2—Minimizing the Probability Distribution Distance
4.1 An Approximation Scheme Under Total Variation Distance
4.2 A One-Pass Approximation Algorithm
4.3 Comparing the Resulting Sequences
5 Conclusions
References
Changing the Uniform Spectrum by Deleting Edges
1 Introduction
2 Complete Graphs
3 The Uniform Spectrum of G + K1
4 Wn Minus Rim Edges
5 Wheel Minus Spokes
6 Problems and Future Work
References
Strongly Regular Multigraphs
1 Introduction
2 Multigraph Considerations
3 Designs
4 Implications for Multigraphs
5 Future Work
References
Geodesic Leech Graphs
1 Introduction
2 Infinite Classes of Geodesic Leech Graphs
3 Concluding Remarks and Open Problems
References
Characterizing s-Strongly Chordal Graphs Using 2-Pathsand k-Chords
1 Introduction
2 s-Strongly Chordal Graphs and 2-Paths
3 Characterizing s-Strongly Chordal Graphs
References
The Spectrum Problem for the 4-Uniform 4-Colorable 3-Cycles with Maximum Degree 2
1 Introduction
1.1 Additional Notation and Terminology
2 Some Small Examples
3 Main Constructions
References
A Matrix Criterion for Harmonic Morphisms of Graphs with Applications to Graph Products
1 Introduction
2 Harmonic Morphisms
2.1 Graph Morphisms and Homomorphisms
2.2 Vertical and Horizontal Multiplicities
2.3 Definition and Properties of Harmonic Morphisms
3 Matrix Criteria for Harmonic Morphisms
3.1 Fundamental Matrix Product Lemmas
3.2 Matrix Criterion for Harmonic Morphisms
3.3 Degree of a Harmonic Morphism
4 Cover Generalizations and Spectra
4.1 Covers and Generalizations
4.2 Spectra
5 NEPS Graphs
5.1 Definition and Background Material
5.2 Projection from a NEPS Graph to a Factor
5.3 Harmonic Morphisms of NEPS Graphs
6 Standard Graph Products
6.1 Tensor Product of Graphs
6.2 Cartesian Product of Graphs
6.3 Strong Product of Graphs
6.4 Lexicographic Product of Graphs
6.5 Projection of Lexicographic Products
7 Examples and a Conjecture
7.1 Blow-Up of Graph
7.2 Join of Graphs
7.3 Projections from Tensor and Cartesian Products
7.4 Grid Graphs
References
Applications of Topological Graph Theory to 2-Manifold Learning
1 Introduction
2 Background Information in Graph Theory and Topological Graph Theory
3 Some Necessary Results from Manifold Learning
4 The Algorithm
4.1 Constructing the 2-Complex
4.2 Building the Rotation System
4.3 Computing Euler Characteristic and Classifying the Resulting Surface
5 Experimental Data and Closing Remarks
5.1 Experimental Data
5.2 Surfaces with Boundary
5.3 On the Importance of Adding Angles That Are Not Too Small
5.4 Topics for Further Investigation
6 Declarations
References
Properties of Sierpinski Triangle Graphs
1 Introduction
2 Sierpinski Triangle Graphs
3 Hanoi Graphs
References
Decomposing the λ-Fold Complete 3-Uniform Hypergraph into the Lines of the Pasch Configuration
1 Introduction
1.1 Additional Notation and Terminology
2 Some Small Examples
3 Main Results
References
On Decompositions of the Johnson Graph
1 Introduction
2 Main Theorem
3 Applications
3.1 Cycle Decompositions
3.2 Path Decompositions
3.3 Decompositions into Other Common Subgraphs
3.4 Decompositions into Two or More Subgraphs
4 Conclusion
References
Tiling With Three Element Sets
1 Introduction
2 Definitions
3 The Greedy Rule
4 The Length of Interval Tilings
5 Tilings in R
5.1 Tiling with Translates of a Single Prototile
5.2 Definitions in the Real Case
5.3 The Greedy Rule
5.4 Interval Tilings
6 Further Results and Questions Regarding Dab
References
Deques on a Torus
1 Introduction
2 Background
3 Toroidal Deques
4 Edge Bounds
5 Cartesian Products
References
Möbius Book Embeddings
1 Background and Definitions
2 Möbius Books
3 An Application
4 An Edge Bound
5 Other Graphs With Möbius Book Thickness One
6 Future Work
References
DNA Self-assembly: Friendship Graphs
1 Introduction
2 Definitions and Prior Results
2.1 Scenario 1
2.2 Scenario 2
2.3 Scenario 3
2.4 Friendship and Generalized Friendship Graphs
3 Friendship Graphs
3.1 Scenario 1
3.2 Scenario 2
3.3 Scenario 3
4 Generalized Friendship Graphs GF4,n
4.1 Scenario 1
4.2 Scenario 2
4.3 Scenario 3
5 Generalized Friendship Graphs GFm,n for m≥5
6 Dumbbell-Friendship Graphs with n C3 Cycles on Each Side
6.1 Scenario 1
6.2 Scenario 2
6.3 Scenario 3
7 Dumbbell-Friendship Graphs with n C4 Cycles on Each Side
7.1 Scenario 1
7.2 Scenario 2
7.3 Scenario 3
8 Conclusion
References
The Pansophy of Semi Directed Graphs
1 Introduction
2 Definition of Pansophy
3 Pansophy for Graphs with One Directed Edge
4 Path Graphs with One Directed Edge
5 Pansophy with Two Edges Directed
6 Conclusions and Open Questions
References
Signed Magic Arrays with Certain Property
1 Introduction
2 The Case m and n Are Even
3 The Case m Odd and n Even
Appendix 1: An Example for Lemma 2 Case 1
Appendix 2: An Example for Lemma 2 Case 2
References
The Buratti-Horak-Rosa Conjecture Holds for Some Underlying Sets of Size Three
1 Introduction
2 Growable Realizations
3 Underlying Sets with Largest Element 5
4 Underlying Set { 1,2, x }
5 An Algorithm
6 Concluding Remarks
References

Citation preview

Springer Proceedings in Mathematics & Statistics

Frederick Hoffman   Editor-in-Chief Sarah Heuss Richard Low John C. Wierman   Editors

Combinatorics, Graph Theory and Computing SEICCGTC 2022, Boca Raton, USA, March 7–11

Springer Proceedings in Mathematics & Statistics Volume 462

This book series features volumes composed of selected contributions from workshops and conferences in all areas of current research in mathematics and statistics, including data science, operations research and optimization. In addition to an overall evaluation of the interest, scientific quality, and timeliness of each proposal at the hands of the publisher, individual contributions are all refereed to the high quality standards of leading journals in the field. Thus, this series provides the research community with well-edited, authoritative reports on developments in the most exciting areas of mathematical and statistical research today.

Frederick Hoffman Editor-in-Chief

Sarah Heuss • Richard Low • John C. Wierman Editors

Combinatorics, Graph Theory and Computing SEICCGTC 2022, Boca Raton, USA, March 7–11

Editor-in-Chief Frederick Hoffman Department of Mathematical Sciences Florida Atlantic University Boca Raton, FL, USA

Editors Sarah Heuss Spartanburg, SC, USA

Richard Low San Jose State University San Jose, CA, USA

John C. Wierman Johns Hopkins University Baltimore, MD, USA

ISSN 2194-1009 ISSN 2194-1017 (electronic) Springer Proceedings in Mathematics & Statistics ISBN 978-3-031-62165-9 ISBN 978-3-031-62166-6 (eBook) https://doi.org/10.1007/978-3-031-62166-6 Mathematics Subject Classification: 05-XX, 12-XX, 12-06, 68RXX, 68R05, 52-XX, 52-06, 05-06, 05CXX © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland If disposing of this product, please recycle the paper.

Preface

The Southeastern International Conference on Combinatorics, Graph Theory, and Computing (SEICCGTC) is an international meeting of mathematical scientists, held annually in March during Spring Break at Florida Atlantic University (FAU) in Boca Raton, Florida. The conference includes a program with plenary lectures by invited speakers, as well as contributed papers by participants. A valuable part of the conference is the opportunity afforded for informal conversations about the methods participants employ in their professional work in business, industry, and government. Conference proceedings appear in the Springer Proceedings in Mathematics & Statistics series. The 53rd meeting of the SEICCGTC was held in the Student Union Building and Live Oak Pavilion on the Boca Raton campus of Florida Atlantic University, March 7–11, 2022. This year, we organized a “hybrid” meeting with a total of 260 people in attendance. One hundred and thirteen people attended “in-person,” while 147 people attended virtually on the Whova platform. A few people attended part of the conference on site, and part of it from another location. A total of 236 participants downloaded and enjoyed some, if not all, of the conference on the Whova platform. The majority of the contributed talks, 112, were presented virtually, while 51 were given in person. The five invited plenary speakers had all intended to participate in person, but two of them had to switch to virtual participation for personal reasons. Conference participants were welcomed by Dr. Teresa Wilcox, Interim Dean of the Charles E. Schmidt College of Science, and by Dr. Stephen Locke, Chair of FAU’s Department of Mathematical Sciences. Invited plenary speakers were Dr. Béla Bajnok of Gettysburg College, Dr. Steve Butler of Iowa State University, Dr. Peter Cameron of the University of Saint Andrews, Dr. Karen Collins of Wesleyan University, and Dr. Jennifer Quinn of the University of Washington. Dr. Bajnok and Dr. Cameron appeared virtually; the others appeared in person. Dr. Collins presented one lecture, and each of the other invited speakers presented two plenary lectures. This year’s special sessions were on “Graph Reconstruction” (organized by Leslie Hogben) and “Matroids and Rigidity Theory” (organized by Daniel Bernstein

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and Zvi Rosen). The conference was expertly coordinated by Dr. Maria Provost. Superb technical support was provided by Andrew Gultz and Jeanne Cimillo. Receptions were held for in-person participants on three evenings, at the Live Oak Patio on Monday and Tuesday evenings, and on Thursday evening at the FAU Club. The conference banquet was held on Wednesday at the Pavilion Grille, a little over a mile from campus. In the history of SEICCGTCs, the 53rd was the first one to be held in hybrid mode. Attendance was either physically present on-site or remote via the Whova virtual conferencing platform. It enjoyed a large number of attendees that totaled 260 and included 76 students. This was a decrease from the 293 that attended the fully virtual 52nd SEICCGTC. A survey was made available in early March 2022 to 260 participants who had attended the 53rd meeting of the SEICCGTC. Seventy-one responses to the online conference satisfaction survey were received for a response rate of 27.3%. Of the total 71 survey respondents, 46% attended virtually, while 25% attended in person, and 12% attended both virtually and in person. Eighty-five percent of those who responded to the survey stated that they would prefer to attend the SEICCGTC 54 “in-person.” The most significant finding of this study is the importance of collegiality among friends and colleagues and collaborators. The conference received partial support from the National Security Agency. Other sponsors included CRCPress/Taylor & Francis, Springer Nature, and The Institute of Combinatorics and its Applications. We feel that the hybrid effort was successful, but we think that an in-person event is more desirable. The members of the editorial board for this volume thank the authors and the referees for their contributions, service, and cooperation. Boca Raton, FL, USA Spartanburg, SC, USA San Jose, CA, USA Baltimore, MD, USA

Frederick Hoffman Sarah Heuss Richard Low John C. Wierman

Volume Overview

Béla Bajnok contributed two chapters based on his plenary lectures. In the first, he leads us on a tour that takes us through four closely related topics: the dual concepts of independence and spanning in finite abelian groups and the analogous dual concepts of designs and distance sets on spheres. He reviews some of the main known results in each area, mentions several open questions, and discusses some connections among the four interesting topics. In his second chapter, Dr. Bajnok discusses some recent results and related open problems in additive combinatorics, with particular attention to sumsets in finite abelian groups. Steve Butler and his coauthor Cailyn Bass develop concepts from the mathematics of juggling. They consider generalizations of drops and descents from permutations to arrangements of sets with repetitions. They establish a generalization of Worpitsky’s identity in the special case when all elements in the set repeat equally often by way of counting passing patterns among jugglers in two different ways. Dinesh G. Sarvate, D. M. Woldemariam, and Li Zhang continue their study of group divisible designs, stressing the difficult cases where the number of groups is less than the block size, and group sizes are different. In this chapter, they consider two sets of cases, building on past work. The results include both constructions and non-existence proofs. Their intention is to help in developing a more unified approach to the construction of such GDDs. Carlos Agrinsoni, Heeralal Janwa, and Moises Delgado obtain an interesting and simple criterion for absolute irreducibility of generalized trinomial multivariable polynomials based on checking squarefreeness of the leading piece and the greatest common divisor of the three pieces. Riordan arrays and the Riordan group go back to 1991. Constructions of the groups and arrays using generating functions often leads to significant combinatorial results. In their chapter, Melkamu Zeleke and Mahendra Jain obtain results for Riordan arrays which lead them to obtain explicit inverses of some elements of the Bell subgroup of the Riordan group. Applications include results on Dyck paths.

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Dennis Davenport, Fatima Fall, Julian Francis, and Trinity Lee obtain some new constructions of production matrices of double Riordan arrays. These can then be used to obtain the A- and Z-sequences for the arrays. Shengwe Lu and Carl Yerger prove the existence of knight’s tours on the surfaces of rectangular surfaces of any size. They present a general algorithm and are able to produce special techniques for the cases where the general algorithm does not apply. John Wierman and Samuel Oberly are able to reduce the upper bound for the site percolation threshold of the square lattice, providing the first improvement since 1995. The substitution method is used, with some computational improvements. A Kirchhoff graph is a vector graph with orthogonal cycles and vertex cuts. Jessica Wang and Joseph D. Fehribach have developed an algorithm that constructs all the Kirchhoff graphs up to a fixed edge multiplicity. They use the algorithm to explore the structure of prime Kirchhoff graph tilings. They establish existence theorems for certain Kirchhoff graphs. A locating set S in a connected graph is a set of vertices such that .N (u) ∩ S is unique for each vertex not in . S. A locating set can be considered as a set of sensors which can determine the exact location of an intruder. The size of a smallest locating set of a graph G is called the locating number of the graph and is denoted by ln(.G). Sul-young Choi and Puhua Guan show that .min{ln (G) : G is a connected graph with n vertices} = n when .2s−1 + (s − 1) < n ≤ 2s + s. Consider a network modeled by a graph G on .n nodes and e edges. There exist several parameters to determine the vulnerability of .G. Alexis Doucette and Charles Suffel introduce a new parameter, j -multiple, k-component order neighbor (k) connectivity, denoted .κn,c,j , which extends multiple domination and .k-component order network connectivity. It is defined as the minimum number of nodes that need to be removed, along with their neighbors, such that the surviving subgraph contains only components of order at most k and every node outside of the set that is adjacent to it is adjacent to at least j nodes from it. The problem of computing this parameter is NP-hard. The authors introduce the parameter, establish its bounds, and compare it to several other previously defined parameters. They also establish formulas for the parameter for several classes of graphs. A variety of network modeling problems begin by generating a degree sequence drawn from a given probability distribution. Brian Cloteaux gives two approaches for generating a graphic approximation in the cases where the sequence is not itself graphic. These schemes are fast, simple to implement, and only require a linear amount of memory. This allows approximation to be performed on very large sequences. Drake Olejniczak and Robert Vandell investigate lengths of paths between vertices in a connected graph. If G is a connected graph of order .n ≥ 2, 1 ≤ k ≤ n−1, an integer, then G is k-uniformly connected if every pair of distinct vertices in G are connected by a path of length .k. For example, .K4 −e is uniformly 2-connected, but not uniformly 1- or 3-connected. A graph G can only be uniformly k-connected if k is at least the diameter of G. For a graph of order .n, (n-1)-uniformly connected

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is Hamiltonian connected. The set of all positive integers k for which a graph G is uniformly k-connected is called the uniform spectrum of G and is denoted by US(G). By looking at .K4 − e and .K4 , one can see that the addition or deletion of a single edge can impact the uniform spectrum of a graph. In their chapter, the authors begin to look at the impact of removing edges from a graph on its uniform spectrum. The family of strongly regular graphs has a direct connection to structures in algebraic combinatorics. These graphs are defined by four parameters: n, the number of nodes; k, the degree of each node; .a, the number of common neighbors of every adjacent pair of nodes; and c, the number of common neighbors for every nonadjacent pair of nodes. Leah H. Meissner and John C. Saccoman discuss strongly regular multigraphs in order to clarify their properties, especially with regard to combinatorial configurations. If .f : E → {1, 2, 3, . . .} is an edge labeling of the graph G, the weight of a path P is the sum of the labels of the edges in the path. The labeling f is called a geodesic Leech labeling if the set of weights of the geodesic paths for G is {1, 2, 3, . . . , .typ (G)}, where .typ (G), the geodesic path number of G, is the number of geodesic paths in G. A graph that admits a geodesic Leech labeling is a geodesic Leech lattice. In their chapter, Seena Varghese, Aparna Lakshmanan S., and S. Arumugam prove the existence of infinite families of geodesic Leech graphs and also prove that .C5 is not a geodesic Leech graph. Some open problems are included. The 1961 characterization of chordal graphs by G. A. Dirac in terms of simplicial vertices was extended by Chvatal, Rusti, and Sritharan to a natural sequence of “weakly chordal graphs.” Similarly, Terry McKee starts from the same point and characterizes a natural sequence of “classical, strongly chordal, . . . , s-strongly chordal” graph, except now in terms of 2-path graphs. The complete t-uniform hypergraph of order v, denoted .Kv(t) , has a set V with v elements as its vertex set and the set of all t-element subsets of V as its edge set. The six authors of this chapter, including two high school students, define a 4-uniform 3-cycle of maximum degree 2 to be any 4-uniform hypergraph of maximum degree 2 that can be obtained by adding two vertices to each of the three edges in .K3(2) . There are five such 4-uniform hypergraphs up to isomorphism. Two of them have chromatic number 4. The authors give necessary and sufficient conditions for the existence of a decomposition of the complete 4-uniform hypergraph of order v into these 4-colorable 3-cycles. Urakawa, Baker, and Norine developed the notion of a harmonic morphism as a type of graph morphism with similar properties to holomorphic maps of Riemann surfaces. Caroline Melles and David Joyner give a matrix criterion for harmonic graph morphisms which allows them to translate combinatorial questions about harmonic morphisms to linear algebra questions. They give several illustrations of the use of their criterion in applications. Tyrus Berry and Steven Schluchter show that, given a sufficiently large point cloud sampled from an embedded 2-manifold in .R n , they may obtain a global representation of the embedded manifold as a cell complex with vertices given by a representative subset of the point cloud. They apply a known projection-based

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approach from the studies of surface reconstruction and computer graphics. They extend this with the application of techniques from topological graph theory for the purposes of topological identification of the sampled surface. They apply the methods to orientable and non-orientable surfaces. ( ) A Nordhaus-Gaddum theorem states bounds on. p (G) + p G and. p(G) · p(G) { } for some graph parameter .p (G) . Viewing. G, G as a decomposition of .Kn allows Allan Bickle to generalize these theorems to decompositions of .Kn with more than two factors. He determines the sum upper bound for independence number, domination number, edge independence number, maximum degree, edge chromatic number, and clique number. He also determines the extremal decompositions for the product lower bound for chromatic number. Let .P denote the 3-uniform hypergraph on 6 vertices whose edges form the lines of the Pasch configuration. Ryan Bunge, Skyler Dodson, Saad El-Zanati, Jacob Franzmeier, and Dru Horne give necessary and sufficient conditions for the existence of P-decompositions of the .λ-fold complete 3-uniform hypergraph on v vertices. Atif Abueida and Mike Daven investigate decompositions of the Johnson graph .J (n, k). The vertices of .J (n, k) are the k element subsets of an n-element set and two vertices are adjacent when they intersect in .k − 1 elements. The authors find conditions for the decompositions of the Johnson graph in a variety of smaller subgraphs, including cycles, paths, and other common subgraphs. Aaron Meyerowitz considers tilings of intervals and other sets of integers with translates of a three-element integer set .A = {0, a, a + b and its reflection .B = {0, b, a + b}. He improves and recasts known results in a way that allows treatment of other cases and similar results. He also considers these questions for intervals and other subsets of the real numbers, with .a, b real. Thomas Mc-Kenzie and Shannon Overbay provide an overview of graphical representations of stacks, queues, and deques. In the case of a cylinder book, corresponding to a deque, they give edge bounds and determine the deque number of the complete graph. They extend the deque in a natural way, forming a toroidal deque. Unlike the other three structures, the toroidal deque can process certain nonplanar graphs, including .K7 and the Cartesian product of two cycles. In their chapter, Nicholas Linthacum, Luke Martin, Thomas McKenzie, Shannon Overbay, and Lin Ai Tan generalize the definition of book embeddings of graphs by allowing book pages to be Mobius strips, rather than half planes. They give an application of these Mobius books and give page bounds for complete graphs and bipartite graphs. They conclude with optimal Mobius book embeddings of some well-known graph families. Based on the flexible tile method for DNA self-assembly, Leyda Almodóvar, Emily Brady, Michaella Fitzgerald, Hsin-Hao Su, and Heiko Todt find a collection of tiles that will construct a nanostructure shaped like a friendship graph or a dumbbell-friendship graph. They find the minimum number tile and bond-edge types required to construct these graphs in three different scenarios representing distinct levels of laboratory constraints.

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Jeffe Boats and Lazaros Kikas consider the question: Given an ordered list of randomly selected pairs of vertices in a graph, how many of these pairs can be connected with disjoint paths? The pansophy of a graph G is the expected number of possible disjoint paths—this has been calculated and studied for many classes of undirected graphs. In this chapter, the authors study the pansophy of various graphs where an edge or a select collection of edges have been directed. They ask how the pansophy is affected and whether specific selections of edges affect pansophy differently from other selections. These and other questions are addressed in the chapter. A signed magic array, .SMA(m, n; s, t), is an .m × n array with the same number of filled cells s in each row and the same number of filled cells t in each column, filled with a certain set of numbers that is symmetric about the number zero, such that every row and column has a zero sum. The notation .SMA(m, n) is used if .m = t; n = s. In their chapter, Abdollah Khodkar and David Leach prove that for every even number .n ≥ 2, there exists an .SMA(m, { } n) such that the entries .±x appear in the same row for every .x ∈ 1, 2, 3, . . . , mn 2 if and only if (a) .n = 2; m ≡ 0, 3 (mod 4) or (b) .n ≥ 4; m ≥ 3. The Buratti-Horak-Rosa Conjecture concerns the possible multisets of edgelabels of a Hamiltonian path in the complete graph with vertex labels 0, 1, 2, . . . , .v − 1 under a particular induced edge-labeling. The conjecture has been proved in a number of cases. Pranit Chand and M. A. Ollis use the method of growable realizations to make further progress on the problem.

Contents

Additive Combinatorics in Groups and Geometric Combinatorics on Spheres. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Béla Bajnok

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A Walk Through Some Newer Parts of Additive Combinatorics . . . . . . . . . . . Béla Bajnok

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Passing Drops and Descents. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Cailyn Bass and Steve Butler

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Group Divisible Designs with Three Groups and Block Size 4. . . . . . . . . . . . . . Dinesh G. Sarvate, Dinkayehu M. Woldemariam, and Li Zhang

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A New Absolute Irreducibility Criterion for Multivariate Polynomials over Finite Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Carlos Agrinsoni, Heeralal Janwa, and Moises Delgado

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On Combinatorial Interpretations of Some Elements of the Riordan Group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Melkamu Zeleke and Mahendra Jani

83

Production Matrices of Double Riordan Arrays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Dennis Davenport, Fatima Fall, Julian Francis, and Trinity Lee

97

The Existence of a Knight’s Tour on the Surface of Rectangular Boxes . . . 111 Shengwei Lu and Carl Yerger A New Upper Bound for the Site Percolation Threshold of the Square Lattice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 John C. Wierman and Samuel P. Oberly Prime, Composite and Fundamental Kirchhoff Graphs. . . . . . . . . . . . . . . . . . . . . 139 Jessica Wang and Joseph D. Fehribach

xiii

xiv

Contents

On the Minimum Locating Number of Graphs with a Given Order . . . . . . . 149 Sul-young Choi and Puhua Guan j -Multiple, k-Component Order Neighbor Connectivity . . . . . . . . . . . . . . . . . . . . 155 Alexis Doucette and Charles Suffel Graphic Approximation of Integer Sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177 Brian Cloteaux Changing the Uniform Spectrum by Deleting Edges . . . . . . . . . . . . . . . . . . . . . . . . . 193 Drake Olejniczak and Robert Vandell Strongly Regular Multigraphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205 Leah H. Meissner and John T. Saccoman Geodesic Leech Graphs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215 Seena Varghese, Aparna Lakshmannan S., and S. Arumugam Characterizing s-Strongly Chordal Graphs Using 2-Paths and k-Chords . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223 Terry A. McKee The Spectrum Problem for the 4-Uniform 4-Colorable 3-Cycles with Maximum Degree 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229 Ryan C. Bunge, Saad I. El-Zanati, Julie N. Kirkpatrick, Shania M. Sanderson, Michael J. Severino, and William F. Turner A Matrix Criterion for Harmonic Morphisms of Graphs with Applications to Graph Products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243 Caroline G. Melles and David Joyner Applications of Topological Graph Theory to 2-Manifold Learning . . . . . . . 275 Tyrus Berry and Steven Schluchter Properties of Sierpinski Triangle Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 297 Allan Bickle Decomposing the .λ-Fold Complete 3-Uniform Hypergraph into the Lines of the Pasch Configuration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 307 Ryan C. Bunge, Skyler R. Dodson, Saad I. El-Zanati, Jacob Franzmeier, and Dru Horne On Decompositions of the Johnson Graph . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 319 Atif Abueida and Mike Daven Tiling With Three Element Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 327 Aaron Meyerowitz Deques on a Torus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 351 Thomas McKenzie and Shannon Overbay

Contents

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Möbius Book Embeddings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 365 Nicholas Linthacum, Luke Martin, Thomas McKenzie, Shannon Overbay, and Lin Ai Tan DNA Self-assembly: Friendship Graphs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 377 Leyda Almodóvar, Emily Brady, Michaela Fitzgerald, Hsin-Hao Su, and Heiko Todt The Pansophy of Semi Directed Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 391 Jeffe Boats and Lazaros Kikas Signed Magic Arrays with Certain Property . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 399 Abdollah Khodkar and David Leach The Buratti-Horak-Rosa Conjecture Holds for Some Underlying Sets of Size Three. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 407 Pranit Chand and M. A. Ollis

Additive Combinatorics in Groups and Geometric Combinatorics on Spheres Béla Bajnok

Abstract We embark on a tour that takes us through four closely related topics: the dual concepts of independence and spanning in finite abelian groups and the analogous dual concepts of designs and distance sets on spheres. We review some of the main known results in each area, mention several open questions, and discuss some connections among these four interesting topics. Keywords Additive combinatorics · Geometric combinatorics

1 Introduction In this chapter we discuss four interrelated topics: two from additive combinatorics, and two from geometric combinatorics. Namely, we introduce the concepts of independence and spanning in the context of finite abelian groups, and we review the seemingly unrelated objects of designs and distance sets on the sphere. As we shall see, strong connections exist among these concepts: independence and spanning are complementary notions in groups, as are designs and distance sets in Euclidean space (or association schemes more generally), and a definite parallel can be established between the former concepts and the latter ones. We will also explain how results from one of these topics can be profitably applied in the development of another.

This chapter is based on the first of two plenary talks given by the author at the 53rd Southeastern International Conference on Combinatorics, Graph Theory & Computing, held at Florida Atlantic University, March 7–11, 2022. B. Bajnok ( □) Department of Mathematics, Gettysburg College, Gettysburg, PA, USA e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 S. Heuss et al. (eds.), Combinatorics, Graph Theory and Computing, Springer Proceedings in Mathematics & Statistics 462, https://doi.org/10.1007/978-3-031-62166-6_1

1

2

B. Bajnok

Our itinerary for visiting these four topics is summarized as follows:

Additive combinatorics in finite abelian groups Geometric combinatorics on the sphere

Maximize t t-independent sets .↓ t-designs

.→

Minimize s s-spanning sets .↑ s-distance sets

We can introduce these four concepts heuristically, as follows. Given a finite subset A in an abelian group G, we measure the degree to which A is independent in G by a nonnegative integer t, and the efficiency with which A generates G by a nonnegative integer s. Analogously, given a finite subset X on the surface of a sphere S, we measure the degree to which X is well balanced on S by a nonnegative integer t, and the number of different distances that X possesses by a nonnegative integer s. We provide precise definitions below.

2 The Independence Number of a Subset of an Abelian Group Throughout, we let G be an additively written abelian group of finite order n. When G is cyclic, we identify it with .Zn = Z/nZ; we consider .0, 1, . . . , n − 1 interchangeably as integers and as elements of .Zn . Our goal is to introduce a notion for the degree to which a subset .A = {a1 , . . . , am } of G is independent in G. Recall that, in an R-module M, a subset .A = {a1 , . . . , am } is linearly independent in M, if the only way to have λ1 · a1 + λ2 · a2 + · · · + λm · am = 0

.

with .λ1 , λ2 , . . . , λm ∈ R is to have λ1 = λ2 = · · · = λm = 0.

.

Clearly, our G is a .Z-module, but finite, so no set A satisfies this. We make the following definition. Definition 1 A subset A is t-independent in G for a nonnegative integer t, if the only way to have λ1 · a1 + λ2 · a2 + · · · + λm · am = 0 Σ with .λ1 , λ2 , . . . , λm ∈ Z and . m i=1 |λi | ≤ t is to have .

λ1 = λ2 = · · · = λm = 0.

.

Additive Combinatorics in Groups and Geometric Combinatorics on Spheres

3

As an example, consider .A = {1, 4, 6, 9, 11} in .G = Z25 . We see that, for instance, .1 + 1 + 1 + 1 − 4, .6 + 6 − 1 − 11, and .1 + 4 + 9 + 11 each equal 0, but no such expression involving fewer than four terms is possible, so A is 3-independent in G. It will be helpful to introduce the following notation: for a positive integer h, we let the h-fold signed sumset of .A = {a1 , a2 , . . . , am } be { } m h± A = λ1 · a1 + · · · + λm · am : λi ∈ Z, Σi=1 |λi | = h .

.

We can thus rephrase Definition 1 to say that A is t-independent in G if .∪th=1 h± A does not contain 0. Returning to our previous example, we find that 1± A = {1, 4, 6, 9, 11, 14, 16, 19, 21, 24}

.

2± A = {2, 3, 5, 7, 8, 10, 12, 13, 15, 17, 18, 20, 22, 23} 3± A = {1, 2, 3, 4, 6, 7, 8, 9, 11, 12, 13, 14, 16, 17, 18, 19, 21, 22, 23, 24} However, .4± A = Z25 , so A is 3-independent but not 4-independent in G. (It is worth noting that our chosen illustration has some special properties; for example, that .0± A, 1± A, and .2± A form an exact partition of the group.) We should point out that independence, as defined above, has strong connections to some classical and well known concepts in additive combinatorics. Namely: • A is a zero-h-sum-free set for some .h ∈ N if x1 + x2 + · · · + xh = 0

.

has no solutions in A. • A is a .(k, 𝓁)-sum-free set for some .k, 𝓁 ∈ N, .k /= 𝓁 if x1 + x2 + · · · + xk = y1 + y2 + · · · + y𝓁

.

has no solutions in A. A .(2, 1)-sum-free set is simply called a sum-free set. • A is a .Bh set for some .h ∈ N if x1 + x2 + · · · + xh = y1 + y2 + · · · + yh

.

has only trivial solutions in A. A .B2 set is called a Sidon set. Observe that A is t-independent in G if, and only if, for every .k, 𝓁 ∈ N0 with k + 𝓁 ≤ t,

.

x1 + x2 + · · · + xk = y1 + y2 + · · · + y𝓁

.

4

B. Bajnok

only has trivial solutions in A (.k = 𝓁 and terms are the same). Therefore, we may employ the classic terminology to restate Definition 1 as follows: Proposition 2 A subset A of G is t-independent in G if, and only if, all of the following hold: • A is zero-h-sum-free for .1 ≤ h ≤ t; • A is .(k, 𝓁)-sum-free for .1 ≤ 𝓁 < k ≤ t − 𝓁; • A is a .Bh set for .2 ≤ h ≤ ⎣ t/2 ⎦ . It is enough, in fact, to require these conditions for equations containing a total of t or .t − 1 terms; therefore the total number of equations considered can be reduced to .2 + (t − 2) + 1 = t + 1. A main question then regarding t-independent sets in abelain groups is to see how large they can get. Problem 3 For each abelian group G and positive integer t, find the size .τ (G, t) of the largest t-independent set in G. Since .A ⊆ G is 1-independent whenever A does not contain 0, we have τ (G, 1) = |G| − 1 for every G; and since .A ⊆ G is 2-independent if, and only if, A is asymmetric (that is, A and .−A are disjoint), we get

.

τ (G, 2) =

.

|G| − |L| , 2

where L is the set of involutions in G. We can evaluate .τ (G, 3) in cyclic groups, as follows. Note that A is 3independent in G if, and only if, it is asymmetric, sumfree, and no three of its (not necessarily distinct) elements add to 0. Therefore, we can observe that the ‘odd’ numbers .1, 3, 5, . . . in .Zn are 3-independent as long as the largest odd element is less than .n/3; in fact, we can go up to just below .n/2 when n is even. We can do better when n has a divisor d that is congruent to 5 mod 6: the union of the first .(d + 1)/6 ‘odd’ cosets of a subgroup of index d form a 3-independent set as well. Using Kneser’s Theorem (cf. [16], [17]), we can prove that we cannot do better: Theorem 4 (Bajnok and Ruzsa, cf. [5]) For any positive integer n we have τ (Zn , 3) |n| ⎧ if n is even; ⎪ 4 ⎪ ⎪ ⎪ ⎪ ⎨ ⎛ ⎞ 1 + p1 n6 if n is odd, has prime divisors 5 mod 6, and p is the smallest; = ⎪ ⎪ ⎪ ⎪ ⎪ |n| ⎩ otherwise. 6

.

A more general result in [5] evaluates .τ (G, 3) for all groups G, with the exception of those whose exponent is a product of a power of 3 and some primes that are congruent to 1 mod 6.

Additive Combinatorics in Groups and Geometric Combinatorics on Spheres

5

For .t = 4, 5, and 6, we have the following computational data: ⎧ ⎪ ⎪ 1 iff ⎪ ⎪ ⎪ ⎨ 2 iff .τ (Zn , 4) = 3 iff ⎪ ⎪ ⎪ 4 iff ⎪ ⎪ ⎩ 5 iff

n ∈ [5, 12] n ∈ [13, 26] n ∈ [27, 45], n = 47 n = 46, n ∈ [48, 68], n = 72, 73 n = 69, 70, 71, n ∈ [74, 102];

⎧ ⎨ 1 iff n ∈ [6, 17], n = 19, 20 τ (Zn , 5) = 2 iff n = 18, n ∈ [21, 37], [39, 41], [43, 45], 47 ⎩ 3 iff n = 38, 42, 46, n ∈ [48, 69], [71, 73], [75, 77], 79, 81, 83, 85, 87; ⎧ ⎨ 1 iff n ∈ [7, 24] τ (Zn , 6) = 2 iff n ∈ [25, 69] ⎩ 3 iff n ∈ [70, 151], [153, 160]. (Here .[k, 𝓁] is the set of integers between k and .𝓁, inclusive.) We can prove the following bounds: Theorem 5 (Bajnok and Ruzsa, cf. [5]) For all .t ≥ 2, .ϵ > 0, and large enough n, we have ⎛ ⎛ | | ⎞ ⎞ 1 1 1 1 t ⎣ t/2 ⎦ ! n ⎣ t/2 ⎦ . . ≤ τ (Zn , t) ≤ −ϵ n · t⎣ (t + 1)/2 ⎦ 2 2 Furthermore, we make the following conjecture. Conjecture 6 Let .t ≥ 2. The value of .

lim

τ (Zn , t) n1/⎣ t/2 ⎦

exists if, and only if, t is even.

3 Spherical Designs Imagine that we want to scatter a finite number .N ∈ N points on a d-dimensional sphere .S d ⊂ Rd+1 : how can we do it in the most “uniformly balanced” way? The answer might be obvious for certain values of N and d: arrange the points to form the vertices of a regular n-gon when .d = 1, or place them so that they form the vertices of a platonic solid when .d = 2 and .N ∈ {4, 6, 8, 12, 20}. But what should we do in general?

6

B. Bajnok

The answer to this question depends, of course, on how we measure the degree to which our pointset .X = {x1 , . . . , xN } ⊂ S d is balanced. For example, we may aim for: • best packing, by maximizing .

min ||xi − xj ||; i/=j

• best covering, by minimizing max min ||x − xi ||;

.

i

x∈S d

• lowest electrostatic energy (Fekete points), by minimizing Σ .

i/=j

1 ; or ||xi − xj ||

• highest thermodynamic entropy (Shub-Smale points), by maximizing ⨅ .

||xi − xj ||.

i/=j

Other well-known criteria include maximizing the volume of the convex hull of X, or minimizing the average probability of error when signal .xi ∈ X is transmitted but (with some probability distribution) some .x ∈ S d is received instead, which then is decoded as the closest .xj ∈ X. But the two concepts that have received the most attention in geometric combinatorics are: • few-distance sets: minimizing the number of distinct distances |{||xi − xj || : i /= j }|; and

.

• spherical designs: maximizing the degree to which the pointset is in momentum balance. In this section we review spherical designs and discuss how independent sets in finite abelian groups (cf. Sect. 2) are related; in Sect. 4 below we return to fewdistance sets and their connections to the other topics of this chapter. To define spherical designs, we assume that .S d is centered at the origin and has radius 1. Given a finite pointset .X ⊂ S d and a polynomial .f : S d → R, we define the average of f over X as fX =

.

1 Σ · f (x); |X| x∈X

Additive Combinatorics in Groups and Geometric Combinatorics on Spheres

7

the average of f over the entire sphere is given by 1 · |S d |

f Sd =

.

∫ f (x)dx Sd

(here .|S d | denotes the surface area of .S d ). We can then make the following definition. Definition 7 (Delsarte, Goethals, and Seidel, cf. [11]) We say that a finite pointset X ⊂ S d is a spherical t-design on .S d for some nonnegative integer t if

.

f X = f Sd

.

holds for every polynomial .f : S d → R of degree at most t. Clearly, any finite pointset .X ⊂ S d is a spherical 0-design. Spherical 1-designs and 2-designs have physical interpretations: it is easy to check that X is a spherical 1-design exactly when X is in mass balance (that is, its center of gravity is at the center of the sphere), and that X is a spherical 2-design if, and only if, X is in both mass balance and inertia balance. One can also verify that the vertices of a regular N-gon form a t-design on .S 1 whenever .N ≥ t + 1, and that the vertices of a regular tetrahedron, octahedron, and icosahedron from a t-design on .S 2 for .t = 2, 3, and 5, respectively. There are numerous other famous examples, such as the 24 vertices of Neil Sloane’s ‘improved snub cube’ that form a 7-design on .S 2 . The general problem is the following. Problem 8 Find all positive integers .d, t, and N for which there is a t-design of size N on .S d . There have been a number of different methods to construct spherical designs, including group theory, numerical analysis, and other techniques; for a sampling of these see [1, 2, 6, 7, 11, 11, 12, 12, 13, 13, 14, 14, 15, 15, 19, 20], and [21]. Here we present a construction using additive combinatorics. By Definition 7, to verify that a given pointset X is a spherical t-design, one should check whether .f X agrees with .f S d for every polynomial f of degree at most t. There is a well-known shortcut: it suffices to do this for non-constant homogeneous harmonic polynomials of degree at most t. (Recall that a polynomial is called harmonic if it satisfies the Laplace equation.) The reduction to homogeneous harmonic polynomials has two advantages. First, there are a lot fewer of them: the set .Harmk (S d ) of homogeneous harmonic polynomials over .S d of degree k forms a vector space over .R whose dimension is only .

dim Harmk (S d ) =

⎛ ⎞ ⎛ ⎞ d +k d +k−2 − . d d

Second, the average of non-constant harmonic polynomials over the sphere is zero. Therefore, we have the following equivalent definition.

8

B. Bajnok

Proposition 9 .X ⊂ S d is a spherical t-design if, and only if, Σ .

f (x) = 0

x∈X

holds for all .f ∈ Harmk (S d ), .k = 1, 2, . . . , t. For small values of k and d, it is possible to construct explicit bases for Harmk (S d ). In particular, for .d = 1 (and arbitrary k) and for .t = 1, 2, 3 (and arbitrary d), we have:

.

Harmk (S 1 ) = 〈{Re(x1 + i · x2 )k , Im(x1 + i · x2 )k }〉

.

Harm1 (S d ) = 〈{xi : 1 ≤ i ≤ d + 1}〉 2 Harm2 (S d ) = 〈{xi2 − xi+1 : 1 ≤ i ≤ d} ∪ {xi xj : 1 ≤ i < j ≤ d + 1}〉

Harm3 (S d ) = 〈{xi3 − 3xi xj2 : 1 ≤ i /= j ≤ d + 1} ∪ {xi xj xk : 1 ≤ i < j < k ≤ d + 1}〉 We now discuss how additive combinatorics—in particular, t-independent sets in the cyclic group .Zn —enables us to construct spherical t-designs explicitly. We first review the case of .d = 1, for which it is well known that the vertices of a regular N-gon form a t-design whenever .N ≥ t + 1. We identify .S 1 with the set { }N of complex numbers of norm 1, and set .X = zj j =1 , where z = e2π i/N = cos (2π/N) + i sin (2π/N) .

.

By PropositionΣ 9, and since .Harmk (S 1 ) = 〈{Re(zk ), Im(zk )}〉 , X is a t-design on .S 1 j k if, and only if, . N j =1 (z ) = 0 for all .k = 1, 2, . . . , t. We see that, when k is not a multiple of N , then .zk /= 1, and thus ( k )N ( N )k N ⎛ ⎞ N ⎛ ⎞ Σ Σ k j −1 −1 z z j k k k =z · = 0. z z . = =z · k k z −1 z −1 j =1

j =1

Since none of .k = 1, 2, . . . , N − 1 is a multiple of N but .k = N is, X is a spherical t-design if, and only if, .N ≥ t + 1, as claimed. (As we will soon see, there are no t-designs on .S 1 of size .N ≤ t.) We now attempt to generalize this construction for higher dimensions. For simplicity, we assume that d is odd, and let .m = (d + 1)/2. (The case when d is even can be reduced to this case by a simple technique, see [2].)

Additive Combinatorics in Groups and Geometric Combinatorics on Spheres

9

Let .a1 , a2 , . . . , am be integers, and set .A = {a1 , a2 , . . . , am }. For a positive integers N, define ⎧ X(A, N) =

.

⎫ ⎞ 1 ⎛ j j j zN (a1 ), zN (a2 ), . . . , zN (am ) | j = 1, 2, . . . , N , √ m

(1)

where j

zN (ai ) = (cos (2πj ai /N ) , sin (2πj ai /N )) .

.

Note that .X(A, N) ⊂ S d . We can then prove the following. Theorem 10 ([2]) Let t, d, and N be positive integers with .t ≤ 3, d odd, and set m = (d + 1)/2. For integers .a1 , a2 , . . . , am , define .X(A, N ) as above. If A is a t-independent set in .ZN , then .X(A, N) is a spherical t-design on .S d .

.

Specifically, from Theorem 10, we see that: • .X(A, N) is a spherical 1-design on .S d if, and only if, A is 1-independent in .ZN . For example, we may take .A = {1} for any .N ≥ 2. (There is no spherical 1-design on .S d with .N = 1.) • .X(A, N) is a spherical 2-design on .S d if, and only if, A is 2-independent in .ZN . For example, we may take .A = {1, 2, . . . , m} for any .N ≥ 2m + 1 = d + 2. (There is no spherical 2-design on .S d with .N ≤ d + 1.) • .X(A, N) is a spherical 3-design on .S d if, and only if, A is 3-independent in .ZN . For example (cf. Sect. 2), we may take: – .A = {1, 3, . . . , 2m − 1} with .N ≥ 6m = 3(d + 1); – .A = {1, 3, . . . , 2m − 1} with .N ≥ 4m = 2(d + 1) if N is even; – .A = any m-subset of .

{pi + 2j + 1 : 0 ≤ i ≤ N/p − 1, 0 ≤ j ≤ (p − 5)/6}

with N≥

.

6p 3p ·m= · (d + 1) p+1 p+1

if N is divisible by p for some .p ≡ 5 mod 6. Therefore, Theorem 10 provides explicit constructions for all possible cases for .t = 1 and .t = 2. For .t = 3, as we will soon see, there is no spherical 3-design on .S d of size .N < 2(d + 1). Our construction yields all cases with even .N ≥ 2(d + 1), and all .N ≥ 52 (d + 1) when N is divisible by 5. In fact, with some additional techniques, we were able to prove the following.

10

B. Bajnok

Theorem 11 ([3]) There is no spherical 3-design on .S d of size .N < 2(d + 1). Spherical 3-designs on .S d exist when: • N is even and .N ≥ 2(d + 1), or • N is odd and .N ≥ 52 (d + 1), but .(d, N ) /= (2, 9), (4, 13). Conjecture 12 Theorem 11 gives all possible .(d, N ) for which a spherical 3design on .S d and of size N exists. Boumova, Boyvalenkov, and Danev in [9] and Boyvalenkov and Stoyanova in [10] provided proofs for some cases of Conjecture 12; in particular they settled all cases with .d < 10, with the exceptions of the possibilities of 9 points forming a spherical 3-design on .S 2 or 13 points forming a spherical 3-design on .S 4 .

4 Spherical Few-Distance Sets In this section we discuss the dual concept to spherical designs: spherical distance sets. (The two topics can also be viewed in the more general setting of P -polynomial and Q-polynomial association schemes; cf. [6].) As before, we let .S d be the unit sphere in .Rd+1 , centered at the origin. For a finite set .X = {x1 , . . . , , xN } of N points on .S d , we let .A(X) be the set of distinct distances A(X) = {||xi − xj || : xi ∈ X, xj ∈ X, i /= j }

.

that they determine. We have the following definition. Definition 13 (Delsarte, Goethals, and Seidel, cf. [11]) We say that a finite pointset .X ⊂ S d is a spherical s-code or spherical s-distance set on .S d for some nonnegative integer s if .|A(X)| = s. For example, it is easy to see that, for any positive integer s, the vertices of a regular N -gon form an s-distance set on .S 1 for .N = 2s + 1, and that the .N = d + 2 vertices of a regular simplex form a 1-distance set on .S d for any positive integer d. There are many other famous and less well known examples: we just mention the regular octahedron and the regular icosahedron on .S 2 , whose vertices form a spherical 2-distance set and a spherical 3-distance set, respectively. The central problem regarding distance sets is the following. Problem 14 Find all positive integers .d, s, and N for which there is an s-distance set of size N on .S d . The following result establishes an upper bound for the size of spherical sdistance sets and a lower bound of the size of spherical t-designs, illustrating nicely

Additive Combinatorics in Groups and Geometric Combinatorics on Spheres

11

the duality of the two concepts. We set A(d, k) =

.

⎛ ⎞ ⎛ ⎞ d + ⎣ k/2 ⎦ d + ⎣ (k − 1)/2 ⎦ + . d d

Theorem 15 (Delsarte, Goethals, and Seidel, cf. [11]) Suppose that .X ⊂ S d has size N , and that it is a spherical t-design as well as a spherical s-distance set on d .S . In this case, A(d, t) ≤ N ≤ A(d, 2s),

.

and thus .t ≤ 2s. And we have the following classification for the case when .t = 2s holds. Theorem 16 (Bannai and Damerell, cf. [7, 8]) Suppose that .X ⊂ S d is a spherical t-design as well as a spherical s-distance set on .S d . If .t = 2s, then .d = 1 or .s ∈ {1, 2}. The cases of .d = 1 (regular polygons) and .s = 1 (regular simplexes) we have already mentioned above. When .s = 2 and .t = 4, then .N = (d 2 + 5d + 4)/2, and we are aware of only two examples (besides the regular pentagon on the circle): the sets corresponding to .(d, N ) = (5, 27) or .(21, 275). The case of .s = 2 has received much attention even when .t < 2s holds. Denoting the maximum size of any spherical 2-distance set on .S d by .M(2, d), we have (d 2 + 5d + 4)/2 ≤ M(2, d) ≤ (d 2 + 3d + 2)/2.

.

The lower bound .Nd = (d 2 + 5d + 4)/2 is given by Theorem 15, while the upper bound .Td = (d 2 + 3d + 2)/2 follows from the fact that the midpoints of the edges of a regular simplex on .S d form a 2-distance set. Musin [18] established the following results. d 1 2 3 4 5 6 − 20 21 22 23 − 38

Nd 5 9 14 20 27 Nd 275 299 Nd

Td M(2, d) 3 5 6 6 10 10 15 16 21 27 Td Td 253 275 276 276 or 277 Td Td

As we see, .M(2, s) agrees with .Nd when .d = 1, 5, or 21, and agrees with .Td for all other d up to 38, except for .d = 4 and possibly .d = 22.

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5 The Spanning Number of a Subset of an Abelian Group We complete our tour through additive and geometric combinatorics by visiting our fourth topic: generating sets. As before, we let G be an abelian group of finite order n, written additively. When G is cyclic, we identify it with .Zn = Z/nZ. Our question here is as follows: how ‘fast’ does a subset of G generate the whole group (if it generates it at all)? We make the following definition. Definition 17 Let .A = {a1 , . . . , am } be an m-subset of G. We say that A is an s-spanning set in G for some nonnegative integer s if every element of G can be written as λ1 · a1 + λ2 · a2 + · · · + λm · am

.

Σ with .λ1 , λ2 , . . . , λm ∈ Z and . m i=1 |λi | ≤ s. Recalling that, for a nonnegative integer h, the h-fold signed sumset of .A = {a1 , a2 , . . . , am } is defined as { } m h± A = λ1 · a1 + · · · + λm · am : λi ∈ Z, Σi=1 |λi | = h ,

.

we can say that A is an s-spanning set of G when .∪sh=0 h± A = G. Let us consider the (especially nice) example of .A = {3, 4} in .G = Z25 . Consider the following illustration.

5 23 1 . 16 19 22 15 18 14

12 8 4 0 21 17 13

11 7 10 3 6 9 24 2 20

In the center, we have .0± A = {0}, the elements nearest to it form .1± A = {3, 4, 21, 22}, the next layer is .2± A = {1, 6, 7, 8, 17, 18, 19, 24}, and finally, the outermost elements form .3± A = {2, 5, 9, 10, 11, 12, 13, 14, 15, 16, 20, 23}. As we can see, .∪3h=0 h± A = G, so A is a 3-spanning set in G. Furthermore, each element of G gets generated exactly once—more on this below. Our main problem regarding spanning sets is the following. Problem 18 For each abelian group G and positive integer s, find the size .φ(G, s) of the smallest s-spanning set in G.

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13

Since .A ⊆ G is 1-spanning if, and only if, .{0} ∪ A ∪ (−A) = G, we have φ(G, 1) =

.

|G| + |L| − 2 , 2

where L is the set of involutions in G. But .φ(G, s) is not known in general for s ≥ 2. For .s = 2, we have the following data in cyclic groups.

.

⎧ ⎪ 1 iff n = 1, 2, 3, 4, 5; ⎪ ⎪ ⎪ ⎪ ⎨ 2 iff n = 6, 7, . . . , 12, 13; .φ(Zn , 2) = 3 iff n = 14, 15, . . . , 21; ⎪ ⎪ ⎪ 4 iff n = 22, 23, . . . , 33, and n = 35; ⎪ ⎪ ⎩ 5 iff n = 34, n = 36, 37, . . . , 49, and n = 51. We can obtain the following bounds. Proposition 19 For all .ϵ, δ, and large enough n, we have .

⎛ √ ⎞ √ √ 1/ 2 − ϵ n ≤ φ(Zn , 2) ≤ (1 + δ) n.

Analogously to how spherical t-designs and s-distance sets are dual concepts, so are t-independent sets and s-generating sets—we explain this next. Let .a(m, s) denote the Delannoy number Σ ⎛s ⎞ ⎛m⎞ · · 2i . .a(m, s) = i i i≥0

(Delannoy numbers may also be defined by the recursion a(m, s) = a(m − 1, s) + a(m − 1, s − 1) + a(m, s − 1),

.

together with the initial conditions of .a(m, 0) = a(0, s) = 1.) We have the following result, paralleling Theorem 15 above. Theorem 20 Suppose that .A ⊂ G, and that it is a t-independent set as well as an s-spanning set in G; assume also that t is even. In this case, a(m, t/2) ≤ |G| ≤ a(m, s),

.

and thus .t ≤ 2s. Cases when equality occurs in Theorem 20 are called perfect; we have the following classification: • .s = 1 (and m arbitrary), in which case .|G| = a(m, 1) = 2m + 1: A is perfect in G if, and only if, A and .−A partition .G \ {0}.

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• .m = 1 (and s arbitrary), in which case .|G| = a(1, s) = 2s + 1: A is perfect in G if, and only if, G is cyclic of order .2s + 1, and .A = {a} with .gcd(a, |G|) = 1. • .m = 2 (and s arbitrary), in which case .|G| = a(2, s) = 2s 2 + 2s + 1: A is perfect in G if G is cyclic of order .2s 2 + 2s + 1 and .A = c · {s, s + 1} with .gcd(c, |G|) = 1 (For example .{3, 4} in .Z25 that we mentioned above). We believe that this rather short list is complete: Conjecture 21 The only instances of perfect sets in G are those three just mentioned. In particular, we must have .s = 1 or .m ∈ {1, 2}. It is worth noting the parallel between Theorem 16 and Conjecture 21. For additional results and open questions on these and other related topics in additive combinatorics, we recommend the author’s book [4].

References 1. B. Bajnok, Construction of spherical t-designs. Geom. Dedicata, 43 (1992), 167–179. 2. B. Bajnok, Constructions of spherical 3-designs. Graphs Combin., 14:2 (1998), 97–107. 3. B. Bajnok, The spanning number and the independence number of a subset of an abelian group. In Number theory (New York, 2003), Springer, New York, 2004, 1–16. 4. B. Bajnok, Additive Combinatorics: A Menu of Research Problems. CRC Press, Boca Raton, 2018, xix+390 pp. 5. B. Bajnok and I. Ruzsa, The independence number of a subset of an abelian group. Integers, 3 (2003), Paper A2. 6. E. Bannai, On extremal finite sets in the sphere and other metric spaces. London Math. Soc. Lecture Note Ser., 131 (1988), 13–38. 7. E. Bannai and R. M. Damerell, Tight spherical designs I. J. Math. Soc. Japan, 31 (1979), 199– 207. 8. E. Bannai and R. M. Damerell. Tight spherical designs II. J. London Math. Soc. (2), 21 (1980), 13–30. 9. S. Boumova, P. Boyvalenkov, and D. Danev, New nonexistence results for spherical designs. In B. Bojanov, editor, Constructive Theory of Functions, Varna, 2002, 225–232. 10. P. Boyvalenkov and M. Stoyanova, New nonexistence results for spherical designs. Adv. Math. Commun. 7 (2013), no. 3, 279–292. 11. P. Delsarte, J. M. Goethals, and J. J. Seidel, Spherical codes and designs. Geom. Dedicata, 6 (1977), 363–388. 12. C. D. Godsil, Algebraic Combinatorics. Chapman and Hall, New York, 1993, xvi+362 pp. 13. J. M. Goethals and J. J. Seidel, Spherical designs. Proc. Sympos. Pure Math., 34 (1979), 255– 272. 14. R. H. Hardin and N. J. A. Sloane, McLaren’s improved snub cube and other new spherical designs in three dimensions. Discrete Comput. Geom., 15 (1996), 429–441. 15. S. G. Hoggar, Spherical t-designs. In C. J. Colbourn and J. H. Dinitz, editors, The CRC handbook of combinatorial designs, CRC Press, Boca Raton, 1996, 462–466. 16. M. Kneser, Abschätzungen der asymptotichen Dichte von Summenmengen. Math. Z. 58 (1953), 459–484. 17. M. B. Nathanson, Additive Number Theory: Inverse Problems and the Geometry of Sumsets. Springer-Verlag, New York, 1996, xiv+293 pp. 18. O. R. Musin, Spherical two-distance sets. J. Combin. Theory Ser. A 116 (2009), no. 4, 988–995.

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19. B. Reznick, Sums of even powers of real linear forms. Mem. Amer. Math. Soc., 463, 1992. 20. J. J. Seidel, Designs and approximation. Contemp. Math., 111 (1990), 179–186. 21. P. D. Seymour and T. Zaslavsky, Averaging sets: A generalization of mean values and spherical designs. Adv. Math., 52 (1984), 213–240.

A Walk Through Some Newer Parts of Additive Combinatorics Béla Bajnok

Abstract In this survey chapter we discuss some recent results and related open questions in additive combinatorics, in particular, questions about sumsets in finite abelian groups. Keywords Additive combinatorics · Sumsets · Finite abelian groups

1 Introduction We embark on a tour through some newer parts of additive combinatorics, visiting some recent results and related open questions. Our context will be within finite abelian groups, written additively. When our group is cyclic and of order n, we identify it with .Zn = Z/nZ; we consider .0, 1, . . . , n−1 interchangeably as integers and as elements of .Zn . Much of additive combinatorics can be described as the study of combinatorial properties of sumsets. Given a nonnegative integer h and an m-subset .A = {a1 , . . . , am } of G, we recall the following definitions and notations: • the h-fold sumset of A: { } m hA = λ1 · a1 + · · · + λm · am | λi ∈ N0 , Σi=1 λi = h ;

.

This chapter is based on the second of two plenary talks given by the author at the 53rd Southeastern International Conference on Combinatorics, Graph Theory and Computing, held at Florida Atlantic University, March 7–11, 2022. B. Bajnok (✉) Department of Mathematics, Gettysburg College, Gettysburg, PA, USA e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 S. Heuss et al. (eds.), Combinatorics, Graph Theory and Computing, Springer Proceedings in Mathematics & Statistics 462, https://doi.org/10.1007/978-3-031-62166-6_2

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• the h-fold restricted sumset of A: { } m .hˆA = λ1 · a1 + · · · + λm · am | λi ∈ {0, 1}, Σi=1 λi = h ; • the h-fold signed sumset of A: { } m .h± A = λ1 · a1 + · · · + λm · am | λi ∈ Z, Σi=1 |λi | = h ; • the h-fold restricted signed sumset of A: { } m .h± ˆ A = λ1 · a1 + · · · + λm · am | λi ∈ {0, ±1}, Σi=1 |λi | = h . These four types of sumsets can be illustrated in the following diagram: Repetition of terms is allowed Terms can be added only

hA .|

Terms can be added or subtracted

Terms must be distinct .⊇

⋂

.h± A

.hˆA .|

.⊇

⋂

.h± ˆA

The sizes of these sumsets may vary greatly: for example, the set .A = {2, 3, 5, 7} in Z53 (in recognition of the 53rd conference) has .|3A| = 14, .|3ˆA| = 4, .|3± A| = 39, and .|3±ˆ A| = 23. In the four sections below we discuss our favorite recent open questions about minimum sumset size, perfect bases and spanning, .(k, 𝓁)-sumfree sets, and maximum-size nonbases, respectively. For additional results and open questions on these and other related topics in additive combinatorics, we recommend the author’s book [9].

.

2 Minimum Sumset Size In this section we aim to address the following general questions: Among the subsets of a finite abelian group, all of a same given size, what is the smallest possible size that their sumsets can be? And, conversely, how can one characterize the subsets of the group that achieve this minimum size? Specifically, given a finite abelian group G and positive integers m and h, we introduce the following notations: ρ(G, m, h) = min{|hA| : A ⊆ G, |A| = m};

.

functions .ρˆ(G, m, h), .ρ± (G, m, h), and .ρ ±ˆ (G, m, h) are defined analogously.

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Among these four quantities, .ρ(G, m, h) has the longest history and it is the only one that is fully known at the present time. The story starts with Cauchy’s result from more than two hundred years ago, which was rediscovered by Davenport a century later, so is now called the Cauchy–Davenport Theorem; we state this result using our notation, as follows. Theorem 1 (Cauchy, cf. [19]; Davenport, cf. [20, 21]) For any prime p and positive integer m, we have ρ(Zp , m, 2) = min{p, 2m − 1}.

.

The general value of .ρ(G, m, h), which is what we explain next, has only been known for about a decade and a half. To start the discussion, we ask: when do subsets A of G have small sumsets? There are two ideas that come to mind. • Put A in a coset of a subgroup: Indeed, if A is a subset of .a + H for some subgroup H of G and element a of A, then .hA is a subset of .h · a + H , and thus can have size at most .|H |. • Put A into an arithmetic progression: If A is a subset of .∪k−1 i=0 {a + i · g} for some hk−h .a ∈ A, .g ∈ G, and positive integer k, then hA is a subset of .∪ i=0 {a + i · g}, and so it can have size at most .hk − h + 1. For the case of m-subsets in groups of prime order p, these two ideas yield the upper bound .min{p, hm − h + 1}. In other groups, we may combine these ideas, and place A inside an arithmetic progression of cosets of some subgroup. Let us carry this out more carefully. In order to simplify notations, we assume that A is an m-subset of the cyclic group .G = Zn . First, we choose a subgroup H of G; if H is of order d, then we have .H = ∪d−1 j =0 {j · n/d}. We will use exactly .⎾m/d⏋ cosets of H that form an arithmetic progression; in particular, if .m = cd + k for some integers c and k with .1 ≤ k ≤ d, then we let A consist of the ‘first’ c cosets of H , plus the ‘first’ k elements of the .(c + 1)-st coset: ⋃ c−1 .A = ∪ ∪k−1 i=0 (i + H ) j =0 {c + j · n/d}. Then A has size .m = cd + k, and we have ⋃ hc−1 .hA = ∪ ∪jhk−h i=0 (i + H ) =0 {hc + j · n/d}, and thus hA has size |hA| = min{n, hcd + min{d, hk − h + 1}}

.

= min{n, hcd + d, hcd + hk − h + 1} = min{n, (hc + 1)d, hm − h + 1} = min{n, (h⎾m/d⏋ − h + 1) · d, hm − h + 1}.

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Introducing the notation fd = (h⎾m/d⏋ − h + 1) · d,

.

we can see that .fn = n and .f1 = hm − h + 1, so |hA| = min{fn , fd , f1 }.

.

Therefore, ρ(Zn , m, h) ≤ min{(h⎾m/d⏋ − h + 1) · d : d|n}.

.

Using Kneser’s Theorem (see [37] or [44]), one can prove that we cannot do better, and we have the following result. Theorem 2 (Plagne, cf. [46]) For all finite abelian groups G of order n and integers m and h, we have ρ(G, m, h) = min{(h⎾m/d⏋ − h + 1) · d : d|n}.

.

Having answered the direct problem of finding .ρ(G, m, h), we now turn to the inverse question: what can we say about m-subsets A of G whose h-fold sumset has size .ρ(G, m, h)? The answer in groups of prime order is a consequence of Vosper’s Theorem, and can be stated as follows. Theorem 3 (Vosper, cf. [48, 49]) Suppose that G is of prime order p. Let m and h be positive integers so that .p > hm − h + 1, and suppose that A is an m-subset of G. Then hA has size .hm − h + 1 if, and only if, .h = 1 or A is an arithmetic progression. The characterization in other groups is likely to be hard in its most general form. For instance, while the only type of 6-subset of .Z15 with a twofold sumset of size .ρ(Z15 , 6, 2) = 9 consists of two cosets of the subgroup of order 3, 7-subsets with twofold sumsets of size .ρ(Z15 , 7, 2) = 13 may come in a variety of forms: unions of two cosets of the subgroup of order 3 plus one element, a coset of the subgroup of order 5 plus two other elements, or an arithmetic progression of size 7. As a modest generalization of Theorem 3, we consider here the case when p is the smallest prime divisor of the order of G, and .m ≤ p. Note that, in this case, from Theorem 2 we have ρ(G, m, h) = min{p, hm − h + 1}.

.

It is not hard to see that the following results follow from Kemperman’s Theorem (cf. [36]) and Kneser’s Theorem, respectively.

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Theorem 4 Let p be the smallest prime divisor of the order of G. Let m and h be positive integers so that .p > hm − h + 1, and suppose that A is an m-subset of G. Then hA has size .hm−h+1 if, and only if, .h = 1 or A is an arithmetic progression. Theorem 5 Let p be the smallest prime divisor of the order of G. Let m and h be positive integers so that .m ≤ p < hm − h + 1, and suppose that A is an m-subset of G. Then hA has size p if, and only if, A is contained in a coset of some subgroup H of G with .|H | = p. The general inverse question remains largely open. Problem 6 For each abelian group G and positive integers m and h, find a characterization of m-subsets of G that have h-fold sumsets of size .ρ(G, m, h). We now turn to minimum sumset size for restricted addition. The value of ρˆ(G, m, h) = min{|hˆA| : A ⊆ G, |A| = m}

.

is largely unknown. However, after several decades of being known as the Erd˝os– Heilbronn Conjecture, the value is finally known in groups of prime order. Theorem 7 (Dias da Silva and Hamidoune, cf. [23]; Alon, Nathanson, and Ruzsa, cf. [1, 2]) For all primes p and positive integers m and h, we have ρˆ(Zp , m, h) = min{p, hm − h2 + 1}.

.

Regarding groups of composite order, we can provide the following sharp result. Theorem 8 ([4]) For all positive integers n, m, and h we have ⎧ min{ρ(Zn , m, 2), 2m − 4} if 2|n and 2|m, ⎪ ⎪ ⎨ or (2m − 2)|n and m − 1 /= 2k ; .ρˆ(Zn , m, 2) ≤ ⎪ ⎪ ⎩ min{ρ(Zn , m, 2), 2m − 3} otherwise. We believe that .ρˆ(Zn , m, 2) is given exactly in Theorem 8, so we ask the following. Problem 9 Prove that equality holds in Theorem 8. Regarding general abelian groups, we mention the following conjecture. Conjecture 10 (Lev, cf. [39]) For any abelian group G and for all positive integers m, we have ρˆ(G, m, 2) ≥ min{ρ(G, m, 2), 2m − 2 − |L|}

.

where L is the subgroup of involutions in G.

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We may observe that in a cyclic group of order n we have .|L| = 2 when n is even and .|L| = 1 when n is odd, cf. Theorem 8 above. We also mention here that Conjecture 10 was established for elementary abelian groups: Theorem 11 (Eliahou and Kervaire, cf. [24]) For all positive integers m and r and for odd primes p, we have ρˆ(Zrp , m, 2) ≥ min{ρ(Zrp , m, 2), 2m − 3}.

.

Note that since .2A = 2ˆA ∪ {0} for any subset A of .Zr2 , we trivially have ρˆ(Zr2 , m, 2) = ρ(Zr2 , m, 2) − 1.

.

The general problem of finding .ρˆ(G, m, h) is largely open and likely to be difficult: as we see, unlike .ρ(G, m, h), it depends on the structure of G and not just the order of G. As a potential next case, we have the following. Conjecture 12 Let p be the smallest prime divisor of the order of G. If m and h are positive integers so that .m ≤ p, then ρˆ(G, m, h) = min{p, hm − h2 + 1}.

.

The inverse problems paralleling Theorems 4 and 5 above are both conjectures for restricted addition. Conjecture 13 Let p be the smallest prime divisor of the order of G. Let m and h be positive integers so that .p > hm − h2 + 1, and suppose that A is an m-subset of G. Then hA has size .hm − h2 + 1 if, and only if, .h = 1; A is an arithmetic progression; or .h = 2, .m = 4, and .A = {a, a + g1 , a + g2 , a + g1 + g2 } for some .a ∈ A and .g1 , g2 ∈ G. Conjecture 14 Let p be the smallest prime divisor of the order of G. Let m and h be positive integers so that .m ≤ p < hm − h2 + 1, and suppose that A is an m-subset of G. Then hA has size p if, and only if, A is contained in a coset of some subgroup H of G with .|H | = p. Károlyi proved that Conjectures 12 and 13 hold for .h = 2, cf. [33–35]. Let us now proceed to signed sumsets and the question of finding ρ± (G, m, h) = min{|h± A| : A ⊆ G, |A| = m}.

.

The value of this function is not generally known, though we have some interesting partial results, and even a conjecture for all cases. Like we have just seen with .ρˆ(G, m, h), the value of .ρ± (G, m, h) depends on the structure of G, not just the order of G. We should mention that, perhaps surprisingly, .ρ± (G, m, h) often agrees with .ρ(G, m, h), although generally .h± A is a lot larger than hA. For example, among groups of order 24 or less, we have .ρ± (G, m, h) = ρ(G, m, h), with the

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only exception of .G = Z23 , .m = 4, and .h = 2, where .ρ± (Z23 , 4, 2) = 8 but 2 .ρ(Z , 4, 2) = 7. The two functions always agree for cyclic groups, though: 3 Theorem 15 (Bajnok and Matzke, cf. [11]) For any cyclic group G and for all positive integers m and h, we have ρ± (G, m, h) = ρ(G, m, h).

.

For the proof of Theorem 15, we constructed a subset .R ⊆ G so that R is symmetric (that is, .R = −R and thus .h± R = hR), has size at least m, and |h± R| ≤ (h⎾m/d⏋ − h + 1) · d.

.

The result now follows from Theorem 2. For noncyclic groups, the situation is substantially more complicated, though [11] contains a conjecture for .ρ± (G, m, h) in all cases. Here we only discuss elementary abelian groups .Zrp ; since we clearly have ρ± (Zr2 , m, h) = ρ(Zr2 , m, h),

.

we assume that p is an odd prime. We have the following rather delicate result. Theorem 16 (Bajnok and Matzke, cf. [12]) Suppose that p is an odd prime and that m, h, and r are positive integers; we use the notation .f1 = hm − h + 1. We define .δ to be 0 or 1, depending on whether .p − 1 is divisible by h or not. We then let k to be the largest integer for which .pk + δ is at most .f1 , and then set c to be the largest integer for which .(hc + 1) · pk + δ is at most .f1 . If .m ≤ (c + 1) · pk , then ρ± (Zrp , m, h) = ρ(Zrp , m, h).

.

We believe that the condition for equality in Theorem 16 is also necessary. Conjecture 17 If, using the notations of Theorem 16, .m > (c + 1) · pk , then ρ± (Zrp , m, h) > ρ(Zrp , m, h).

.

Using results of Vosper, Kemperman, and Lev, we have proved Conjecture 17 for the case of .r = 2 and .h = 2. The inverse problems regarding .ρ± (G, m, h) are quite interesting as well: perhaps surprisingly, symmetric subsets don’t always yield minimum signed sumset size. Recall that a subset A in a group is symmetric if A equals .−A and asymmetric when A and .−A are disjoint. In addition, we define A to be near-symmetric when it is possible to remove one element from it after which it becomes symmetric. We have the following inverse result.

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Theorem 18 (Bajnok and Matzke, cf. [11]) Suppose that G is a finite abelian group and that m and h are positive integers. Let .A(G, m) denote the collection of m-subsets of g that are symmetric, asymmetric, or near-symmetric. We then have ρ± (G, m, h) = min{|h± A| : A ∈ A(G, m)}.

.

We note that all three types of subsets are essential; it may be an interesting question to characterize all situations where the minimum signed sumset size is achieved by symmetric, asymmetric, or near-symmetric subsets, respectively. We close this section by mentioning our fourth quantity, ρ ±ˆ (G, m, h) = min{|hA| : A ⊆ G, |A| = m},

.

though only to say that we know very little about it. Clearly, ρˆ(G, m, h) ≤ ρ ±ˆ (G, m, h) ≤ ρ± (G, m, h),

.

and both inequalities may be strict. Perhaps a good place to start the evaluation is in cyclic groups and for .h = 2. Problem 19 Evaluate .ρ ±ˆ (Zn , m, 2) for positive integers m and n.

3 Perfect Bases and Spanning Sets In this section we look for perfection: subsets of abelian groups that generate each element of the group uniquely. Namely, for a finite abelian group G, a subset A of G, and a positive integer s we introduce the following definitions: • A is a perfect s-basis in G if each element of G can be written uniquely as a sum of at most s elements of A. • A is a perfect restricted s-basis in G if each element of G can be written uniquely as a sum of at most s distinct elements of A. • A is a perfect s-spanning set in G if each element of G can be written uniquely as a signed sum of at most s elements of A. • A is a perfect restricted s-spanning set in G if each element of G can be written uniquely as a signed sum of at most s distinct elements of A. As we will see, perfection does exist, though sometimes it may be difficult to find. We start with perfect s-bases. As far as we know, perfect bases have not been investigated yet before, though the concept of s-basis has enjoyed a rich history since it was first discussed by Erd˝os and Turán in [29] eighty years ago; see, for example, [26, 28, 38, 42, 44], and [43]. Trivially, the set of nonzero elements is a perfect 1-basis in any group G, and the 1-element set consisting of a generator of G is a perfect s-basis in the cyclic group of order .s + 1. It turns out that there are no others:

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25

Theorem 20 (Bajnok, Berson and Just, cf. [10]) If a subset A of a finite abelian group G is a perfect s-basis, then .s = 1 and .A = G \ {0}, or G is cyclic of order .s + 1 and A consists of a single element. The proof of Theorem 20 is based on the fact that if A is a perfect s-basis in G, then the .s + 1 subsets .

− A, A − A, 2A − A, . . . , (s − 1)A − A, and (s − 1)A

are pairwise disjoint. We can then compute that ⎛

s−1 .| ∪ h=0

⎞ ⎛ ⎞ m+s (m − 1)(s − 1) m + s − 2 (hA − A) ∪ (s − 1)A| = + . s s s−1

However, if A is a perfect s-basis of size m in G, then |G| =

s ⎲

.

|hA| =

h=0

⎞ s ⎛ ⎲ m+h−1 h=0

h

⎛ =

⎞ m+s , s

and this is only possible if .m = 1 or .s = 1, as claimed. Moving on to perfect restricted s-bases, we first observe that the instances of .|A| = 1 or .s = 1 are identical for restricted addition and unrestricted addition, and are as listed in Theorem 20. As a major contrast, however, there are infinitely many perfect restricted s-bases with .|A| ≥ 2 and .s ≥ 2, as we explain below. We are able to provide a complete characterization for the case of .s = 2, for which we have the following result. Theorem 21 (Bajnok, Berson, and Just, cf. [10]) A finite abelian group G has a perfect restricted 2-basis if, and only if, it is isomorphic to one of the following groups: .Z2 , Z4 , Z7 , Z22 , Z42 , or .Z22 × Z4 . Our strategy for proving Theorem 21 is similar to the unrestricted case, but exhibiting sets that are pairwise disjoint seems more elusive, hence our result is only for .s = 2. For a given perfect restricted 2-basis A, we consider the set T = (A − A) ∪ A,

.

and prove that, unless G isomorphic to .Z4 , .Z7 , or .Z22 × Z4 (in which cases perfect restricted 2-bases exist) or to an elementary abelian 2-group, T has size more than the order of G. However, if all nonzero elements of G have order 2, then A − A = 2A = {0} ∪ 2ˆA,

.

and thus .T = ∪2h=0 hˆA. Therefore, A being a perfect restricted 2-basis in G is equivalent to having .T = G, resulting in no contradiction.

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Luckily, a problem of Ramanujan comes to the rescue. In 1913, Ramanujan asked in [47] whether the quantity .2k − 7 can be a square number for any integer k besides 3, 4, 5, 7, and 15 (see also Question 464 in [17]). The negative answer was given by Nagell in 1948 (see [40]; [41] for the English version). Suppose now that G is the elementary abelian 2-group of rank r that has a perfect restricted 2-basis of size m: we then must have ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ m m m r .2 = + + , 0 1 2 and therefore 2r+3 − 7 = (2m + 1)2 .

.

We thus have exactly four choices for r: 1, 2, 4, and 12. Perfect restricted 2-bases exist for the first three choices, and we were able to show that they do not in .Z12 2 , completing the proof of Theorem 21. We know of perfect restricted s-bases of size m exist for higher s as well: • for .m ≤ s, in every group of order .|G| = 2m ; • for .m = s + 1, in .G ∼ = Z2s+1 −1 ; and ∼ 2s−2 × Z4 . • for .m = 2s + 1, in .G ∼ = Z2s 2 , as well as in .G = Z We believe that there are no others. Conjecture 22 For integers .m ≥ 2 and .s ≥ 2, perfect restricted s-bases of size m exist if, and only if, G has order .2m or is isomorphic to .Z2s+1 −1 , .Z2s 2 , or to 2s−2 × Z . .Z 4 Next, we turn to perfect spanning sets, that is, subsets A of our group G where each element of G can be written uniquely as a signed sum of at most s elements of A. Note that, if A is a perfect s-spanning set of G and is of size m, then the order of G must equal the Delannoy number ⎲ ⎛s ⎞ ⎛m⎞ · · 2i . .a(m, s) = i i i≥0

(Delannoy numbers may also be defined by the recursion a(m, s) = a(m − 1, s) + a(m − 1, s − 1) + a(m, s − 1),

.

together with the initial conditions of .a(m, 0) = a(0, s) = 1.) We are aware of only the following perfect spanning sets: • .s = 1 (and m arbitrary), in which case .|G| = a(m, 1) = 2m + 1: A is perfect in G if, and only if, A and .−A partition .G \ {0}. • .m = 1 (and s arbitrary), in which case .|G| = a(1, s) = 2s + 1: A is perfect in G if, and only if, G is cyclic of order .2s + 1, and .A = {a} with .gcd(a, |G|) = 1.

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27

• .m = 2 (and s arbitrary), in which case .|G| = a(2, s) = 2s 2 + 2s + 1: A is perfect in G if G is cyclic of order .2s 2 + 2s + 1 and .A = c · {s, s + 1} with .gcd(c, |G|) = 1. We believe that this rather short list is complete: Conjecture 23 The only instances of perfect sets in G are those three just mentioned. In particular, we must have .s = 1 or .m ∈ {1, 2}. At the present time, we know little about perfect restricted spanning sets, though we suspect that their theory (with the powers of 3 replacing the powers of 2) can be developed analogously to perfect restricted bases.

4 (k, l)-Sumfree Sets Recall that a subset A of G is called sumfree if it does not contain the sum of two (not necessarily distinct) of its elements, that is, if A and 2A are disjoint. More generally, for positive integers k and l, with .k > l, we call a subset A of G .(k, l)sumfree if kA and lA are disjoint. The main question we intend to discuss in this section is how large a .(k, l)-sumfree subset of a given group can be. Problem 24 For each pair of positive integers k and l, and for each finite abelian group G, find the maximum size .μ(G, {k, l}) of .(k, l)-sumfree subsets in G. Sumfree sets in abelian groups were first introduced by Erd˝os in [25] and then studied systematically by Wallis, Street, and Wallis in [50]. We can construct sumfree sets in G by selecting a subgroup H in G for which .G/H is cyclic and then taking the ‘middle one-third’ of the cosets of H . More precisely, with d denoting the index of H in G, we see that ⎧⎾ μ(G, {2, 1}) ≥ max

.

d|e(G)

⏋ ⎫ d −1 n · , 3 d

where .e(G) is the exponent of G. Using a version of Kneser’s Theorem, Diamanda and Yap (see [22] or [50]) proved in the 1960s that we cannot do better in cyclic groups; this result was only extended to the general case in 2005: Theorem 25 (Green and Ruzsa, cf. [31]) For any abelian group G of order n and exponent .e(G), we have ⎧⎾ μ(G, {2, 1}) = max

.

d|e(G)

⏋ ⎫ d −1 n · . 3 d

We should note that the proof of this result relies in part on a computer program. For general k and l, the first result was given by Bier and Chin (cf. [18]) who evaluated .μ(Zp , {k, l}) for prime values of p. This was generalized by Hamidoune

28

B. Bajnok

and Plagne for general cyclic groups, but only when their order was relatively prime to .k − l: Theorem 26 (Hamidoune and Plagne, cf. [32]) If .k − l is relatively prime to n, then ⎧⎾ ⏋ ⎫ d −1 n · . .μ(Zn , {k, l}) = max d|n k+l d Just recently, we were able to generalize Theorem 26 to the case when n and .k −l are not relatively prime. Theorem 27 (Bajnok and Matzke, cf. [13]) For all positive integers n, k, and l, with .k > l, we have ⎧⎾ ⏋ ⎫ d − (δ − r) n · , .μ(Zn , {k, l}) = max d|n k+l d where .δ = gcd(d, k − l) and r is the remainder of .l⎾(d − δ)/(k + l)⏋ mod .δ. Let us review our approach for the proof of Theorem 27. The main role is played by arithmetic progressions, that is, sets of the form A = {a + i · b | i = 0, 1, . . . , m − 1}

.

for some positive integer m and elements a and b of .Zn . In [32], Hamidoune and Plagne proved that, if n and .k − l are relatively prime, then .μ(Zn , {k, l}) equals { n} , . max α(Zd , {k, l}) · d|n d where .α(Zd , {k, l}) is the maximum size of a .(k, l)-sumfree arithmetic progression in .Zd . Although the situation is considerably more intricate when n and .k − l are not relatively prime, it turns out (see [3]) that the identity remains valid. This then reduces the problem of finding .μ(Zn , {k, l}) to arithmetic progressions only. When trying to evaluate .α(Zd , {k, l}), one naturally considers two types of arithmetic progressions: those with a common difference that is not relatively prime to d (in which case the set is contained in a coset of a proper subgroup), and those where the common difference is relatively prime to d (in which case the set, unless of size 1, is not contained in a coset of a proper subgroup). The main difficulty is caused by those of the first type; luckily, however, we were able to prove that it does not matter: Theorem 28 (Bajnok and Matzke, cf. [13]) For all positive integers n, k, and l with .k > l we have { n} , .μ(Zn , {k, l}) = max γ (Zd , {k, l}) · d|n d where .γ (Zd , {k, l}) is the size of the largest .(k, l)-sumfree interval in .Zd .

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29

Since intervals are easy to handle, finding .γ (Zd , {k, l}) is a routine calculation; this then yields Theorem 27 above. Turning now to the inverse problem of classifying all .(k, l)-sumfree subsets of G of maximum size, we first mention that the problem for .(k, l) = (2, 1) has finally been completed in 2016 in a complicated paper by Balasubramanian, G. Prakash, and D. S. Ramana (see [16]). For other k and l, we have the following result. Theorem 29 (Plagne, cf. [45]) Let p be a prime and k and l be positive integers with .k > l and .k ≥ 3; assume also that p does not divide .k − l. If A is a .(k, l)-sumfree set of size | |A| = μ(Zp , {k, l}) =

.

| p−2 +1 k+l

in .Zp , then A is an arithmetic progression. The result is interesting in that it is simpler than the sumfree case (where sumfree subsets that are not arithmetic progressions may also have maximum size .μ(Zp , {2, 1}); see [9]). Problem 30 For all positive integers n, k, and l, classify all .(k, l)-sumfree subsets of .Zn that have maximum size .μ(Zn , {k, l}).

5 Sumset Size of Maximum-Size Nonbases Recall that a nonempty subset A of G is h-complete (alternatively, a basis of order h) if .hA = G and that if hA is a proper subset of G, we say that A is h-incomplete. The h-critical number .χ (G, h) of G is defined as the smallest positive integer m for which all m-subsets of G are h-complete; that is: χ (G, h) = min{m : A ⊆ G, |A| ≥ m ⇒ hA = G}.

.

It is easy to see that for all G and h we have .hG = G, so .χ (G, h) is well defined. The value of .χ (G, h) is now known for every G and h. Theorem 31 ([5]) For all abelian groups G of order n and all positive integers h, we have | ⎞ ⎫ ⎧⎛| n d −2 +1 · : d|n + 1. .χ (G, h) = max h d

30

B. Bajnok

In particular, by Theorem 31, we have the following explicit formulas: χ (G, 1) = n;

.

χ (G, 2) =

|n| 2

+ 1;

⎞ ⎧⎛ ⎪ 1 + p1 n3 if n has prime divisors congruent to 2 mod 3, ⎪ ⎪ ⎨ and p is the smallest such divisor, χ (G, 3) = ⎪ ⎪ ⎪ ⎩|n| otherwise. 3 Explicit expressions for .χ (G, h) get increasingly more complicated as h increases. The question that we here try to address is the following: What can one say about the size of hA if A is an h-incomplete subset of maximum size in G? Namely, we aim to determine the set S(G, h) = {|hA| : A ⊂ G, |A| = χ (G, h) − 1, hA /= G}.

.

Trivially, .S(G, 1) = {n − 1}. For .h = 2 and .h = 3 we have the following results. Theorem 32 (Bajnok and Pach, cf. [14]) Let G be an abelian group of order n. • If the exponent of G is divisible by 4, then S(G, 2) = {n − n/d : d|n, 2|d} .

.

• If the exponent of G is even but not divisible by 4, then S(G, 2) = {n − n/d : d|n, 2|d, d /= 4} .

.

• If n is odd and .n > 9, then .S(G, 2) = {n − 2, n − 1}. Theorem 33 (Bajnok and Pach, cf. [14]) Let G be an abelian group of order n. • If n has prime divisors congruent to 2 mod 3 and p is the smallest, then .S(G, 3) = {n − n/p}. • If n has no prime divisors congruent to 2 mod 3 but .3|n, then S(G, 3) = {n − n/d : d|n, 3|d, d /= 3}

.

∪ {n − 2n/d : d|n, 1 ≤ ν3 (d) ≤ ν3 (e(G))} , where .e(G) is the exponent of G, and .ν3 (t) is the highest power of 3 that divides t.

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31

• If all divisors of n are congruent to 1 mod 3, but .G /∼ = Zr7 , then .S(G, 3) = {n − 3, n − 1}. • If .G ∼ = Zr7 , then .S(G, 3) = {n − 3}. The analogous question for restricted addition is considerably more challenging; in fact, we do not even know the value of the restricted critical number χˆ(G, h) = min{m : A ⊆ G, |A| ≥ m ⇒ hˆA = G}

.

for .h ≥ 3 in all cases. However, for .h = 2 we have .χˆ(G, 2) = ⎿n/2⏌+2 (cf. [5, 7]), and the following result for Sˆ(G, 2) = {|2ˆA| : A ⊂ G, |A| = χˆ(G, 2) − 1, 2ˆA /= G}

.

in the case of cyclic groups. Theorem 34 (Gallardo, Grekos, et al.; cf. [30]) Let G be a cyclic abelian group of order n. • If .n = 2k for some positive integer k, then .Sˆ(G, 2) = {n − 1}. • In all other cases, .Sˆ(G, 2) = {n − 2, n − 1}. The problem of finding critical numbers .χˆ(G, h) for other G and h (cf. [6] for a review of known cases), and the corresponding sets .Sˆ(G, h), appear to be challenging.

References 1. N. Alon, M. B. Nathanson, and I. Ruzsa, Adding distinct congruence classes modulo a prime. Amer. Math. Monthly 102 (1995), no. 3, 250–255. 2. N. Alon, M. B. Nathanson, and I. Ruzsa, The polynomial method and restricted sums of congruence classes. J. Number Theory 56 (1996), no. 2, 404–417. 3. B. Bajnok, On the maximum size of a (k, l)-sumfree subset of an abelian group. Int. J. Number Theory 5 (2009), no. 6, 953–971. 4. B. Bajnok, On the minimum size of restricted sumsets in cyclic groups. Acta Math. Hungar. 148 (2016), no. 1, 228–256. 5. B. Bajnok, The h-critical number of finite abelian groups. Unif. Distrib. Theory 10 (2015), no.2, 93–15. 6. B. Bajnok, More on the h-critical numbers of finite abelian groups. Ann. Univ. Sci. Budapest. Eötvös Sect. Math. 59 (2016), 113–122. 7. B. Bajnok, Corrigendum to “The h-critical number of finite abelian groups.” Unif. Distrib. Theory 12 (2017), no. 2, 119–124. 8. B. Bajnok, Open problems about sumsets in finite abelian groups: minimum sizes and critical numbers. Combinatorial and Additive Number Theory II: CANT, New York, NY, USA, 2015 and 2016, Springer, New York, 2017, 9–23. 9. B. Bajnok, Additive Combinatorics: A Menu of Research Problems. CRC Press, Boca Raton, 2018, xix+390 pp.

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10. B. Bajnok, C. Berson, and H. A. Just, On Perfect Bases in Finite Abelian Groups. Involve 15 (2022), No. 3, 525–536. 11. B. Bajnok and R. Matzke, The minimum size of signed sumsets. Electron. J. Combin. 22 (2015), no. 2, Paper 2.50, 17 pp. 12. B. Bajnok and R. Matzke, On the minimum size of signed sumsets in elementary abelian groups. J. Number Theory 159 (2016), 384–401. 13. B. Bajnok and R. Matzke, On the maximum size of (k, l)-sumfree sets in cyclic groups. Bulletin of the Australian Mathematical Society 99 (2019), no. 2, 184–194. 14. B. Bajnok and P. P. Pach, On sumsets of nonbases of maximum size. To appear in European Journal of Combinatorics. 15. B. Bajnok and I. Ruzsa, The independence number of a subset of an abelian group. Integers 3 (2003), A2, 23 pp. 16. R. Balasubramanian, G. Prakash, and D. S. Ramana, Sum-free subsets of finite abelian groups of type III. European J. Combin. 58 (2016), 181–202. 17. B. C. Berndt, Y-S. Choi, and S-I. Kang, The problems submitted by Ramanujan to the Journal of the Indian Mathematical Society. Contemp. Math. 236 (1999), 15–56. 18. T. Bier and A. Y. M. Chin, On (k, l)-sets in cyclic groups of odd prime order. Bull. Austral. Math. Soc. 63 (2001), no. 1, 115–121. 19. A–L. Cauchy, Recherches sur les nombres. J. École Polytechnique 9 (1813), 99–123. 20. H. Davenport, On the addition of residue classes. J. London Math. Soc. 10 (1935), 30–32. 21. H. Davenport, A historical note. J. London Math. Soc. 22 (1947), 100–101. 22. P. H. Diananda and H. P. Yap, Maximal sumfree sets of elements of finite groups. Proc. Japan Acad. 45 (1969), 1–5. 23. J. A. Dias Da Silva and Y. O. Hamidoune, Cyclic space for Grassmann derivatives and additive theory. Bull. London Math. Soc. 26 (1994), no. 2, 140–146. 24. S. Eliahou and M. Kervaire, Sumsets in vector spaces over finite fields. J. Number Theory 71 (1998), no. 1, 12–39. 25. P. Erd˝os, Extremal problems in number theory. Proc. Sympos. Pure Math., Vol. VIII pp. 181– 189 Amer. Math. Soc., Providence, R.I., 1965. 26. P. Erd˝os and R. L. Graham, On bases with an exact order. Acta Arith. 37 (1980), 201–207. 27. P. Erd˝os and H. Heilbronn, On the addition of residue classes mod p. Acta Arith. 9 (1964), 149–159. 28. P. Erd˝os and M. B. Nathanson, Problems and results on minimal bases in additive number theory. Lecture Notes in Math. 1240, Springer, Berlin (1987), 87–96. 29. P. Erd˝os and P. Turán, On a problem of Sidon in additive number theory and some related questions. J. London Math. Soc. 16 (1941), 212–215. 30. L. Gallardo, G. Grekos, L. Habsieger, F. Hennecart, B. Landreau, and A. Plagne, Restricted addition in Z/nZ and an application to the Erd˝os–Ginzburg–Ziv problem. J. London Math. Soc. (2) 65 (2002), no. 3, 513–523. 31. B. Green and I. Ruzsa, sumfree sets in abelian groups. Israel J. Math. 147 (2005), 157–188. 32. Y. O. Hamidoune and A. Plagne, A new critical pair theorem applied to sumfree sets in abelian groups. Comment. Math. Helv. 79 (2004), no. 1, 183–207. 33. Gy. Károlyi, On restricted set addition in abelian groups. Ann. Univ. Sci. Budapest, Eötvös Sect. Math. 46, (2003) 47–54 (2004). 34. Gy. Károlyi, The Erd˝os–Heilbronn problem in abelian groups. Israel J. Math. 139 (2004) 349– 359. 35. Gy. Károlyi, An inverse theorem for the restricted set addition in abelian groups. J. Algebra 290 (2005), no. 2, 557–593. 36. J. H. B. Kemperman, On small sumsets in an abelian group. Acta Math. 103 (1960), 63–88. 37. M. Kneser, Abschätzungen der asymptotichen Dichte von Summenmengen. Math. Z. 58 (1953). 459–484. 38. V. Lambert, T. H. Lê, and A. Plagne, Additive bases in groups. Israel J. Math. 217 (2017), no. 1, 383–411.

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39. V. F. Lev, Restricted set addition in groups. I. The classical setting. J. London Math. Soc. (2) 62 (2000), no. 1, 27–40. 40. T. Nagell, Løsning til oppgave nr. 2, 1943, s. 29. Nordisk Mat. Tidskr. 30 (1948), 62–64. 41. T. Nagell, The Diophantine Equation x 2 + 7 = 2n . Ark. Mat. 4 (1961), 185–187. 42. M. B. Nathanson, Minimal bases and maximal nonbases in additive number theory. J. Number Theory 6 (1974), 324–333. 43. M. B. Nathanson, Paul Erd˝os and additive bases. arXiv:1401.7598 [math.NT] (2014). 44. M. B. Nathanson, Additive number theory. Inverse problems and the geometry of sumsets. Graduate Texts in Mathematics, 165. Springer–Verlag, New York, 1996. xiv+293 pp. 45. A. Plagne, Maximal (k, l)-free sets in Z/pZ are arithmetic progressions. Bull. Austral. Math. Soc. 65 (2002), no. 3, 137–144. (k) 46. A. Plagne, Optimally small sumsets in groups, I. The supersmall sumset property, the μG and (k) the νG functions. Unif. Distrib. Theory 1 (2006), no. 1, 27–44. 47. S. Ramanujan, Question 464. Journal of the Indian Mathematical Society 5 (1913), 120. 48. A. G. Vosper, The critical pairs of subsets of a group of prime order. J. London Math. Soc. 31 (1956), 200–205. 49. A. G. Vosper, Addendum to “The critical pairs of subsets of a group of prime order”. J. London Math. Soc. 31 (1956), 280–282. 50. W. D. Wallis, A. P. Street, and J. S. Wallis, Combinatorics: room squares, sumfree sets, Hadamard matrices. Lecture Notes in Mathematics, 292, Springer–Verlag, Berlin-New York, 1972. iv+508 pp.

Passing Drops and Descents Cailyn Bass and Steve Butler

Abstract We consider generalizations of drops and descents from permutations to arrangements of sets with repetition. We also establish a generalization of Worpitzky’s identity in the special case when all elements in the set repeat equally often by way of counting passing patterns among jugglers in two different ways. Keywords Worpitzky’s identity · Permutations · Drops · Descents

1 Introduction The paper Juggling drops and descents by Buhler, Graham, Eisenbud, and Wright [3] was one of the first mathematical papers to introduce siteswap notation and used this notation, together with Worpitzky’s identity (see Theorem 1 with .k = 1), to get a count for the number of different patterns that can be juggled with at most b balls. Subsequent work by Ehrenborg and Readdy [10] gave a completely different technique to count the same set of patterns. By combining these two results, we get a combinatorial proof of Worpitzky’s identity. Since these papers have come the mathematical community’s understanding of juggling has continued to grow to topics including random juggling [14, 17], multiplex juggling (where multiple balls can be caught/thrown by a juggler at the same time) [5, 7], passing (involving multiple jugglers with the ability to pass objects between them) [4, 6], and more [1, 8, 9, 13]. Our focus here will be on giving two different interpretations of how to describe and count the number of possible passing patterns, which are juggling patterns

C. Bass School of Mathematical and Statistical Sciences, Clemson University, Clemson, SC, USA e-mail: [email protected] S. Butler (✉) Department of Mathematics, Iowa State University, Ames, IA, USA e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 S. Heuss et al. (eds.), Combinatorics, Graph Theory and Computing, Springer Proceedings in Mathematics & Statistics 462, https://doi.org/10.1007/978-3-031-62166-6_3

35

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C. Bass and S. Butler

involving multiple jugglers. As a consequence we will give a combinatorial proof of the following generalization of Worpitzky’s identity.

Theorem 1 We have ⎛ ⎞n ⎲ / \ ⎛ ⎞ x n x+𝓁 . = k 𝓁 k kn 𝓁

〈 〉 where . n𝓁 k is the number of arrangements of .{1(k) , 2(k) , . . . , n(k) } with .𝓁 descents. (Here .a (b) indicates the element a repeated b times.)

This variant of Worpitzky’s identity is a special case of one given algebraically by 〈 〉 Pita-Ruiz [15, 16]. A combinatorial interpretation of the coefficients . n𝓁 k as well as a combinatorial proof of Theorem 1 was given by Engbers et al. [11]; 〈 〉our approach will give a different combinatorial interpretation of the coefficients . n𝓁 k as well as a different combinatorial proof of Theorem 1 (in the language of permutation statistics our approach is drop-focused instead of descent-focused). We will proceed by first introducing a generalization for the Eulerian numbers in Sect. 2, and establishing a few basic properties of these numbers. We then introduce the basics of passing patterns in Sect. 3, and give a method to describe these patterns corresponding to using cards. While in Sect. 4 we will give a method to describe passing patterns by siteswap which uses a generalization of rook placements. Finally, we can equate the enumeration of the two different descriptions of passing patterns of period n with k jugglers and at most b balls with every juggler catching/throwing a ball at each beat to derive Theorem 1.

2 Generalization of Eulerian Numbers Two well-studied permutation statistics are descents and drops, which are tied to the Eulerian numbers. We now generalize these ideas to arrangements of sets which might involve repeated terms, e.g. .{1(a1 ) , . . . , n(an ) } where the term in the exponent represents how many copies of the value we have. The easier idea to generalize is descents. Given an arrangement .ρ = ρ1 . . . ρm , we let .des(ρ), the number of descents of .ρ, be the number of occurrences where .ρi > ρi+1 for .1 ≤ i ≤ m − 1. To generalize drops we first introduce .τ which is the arrangement of (a ) (a ) .{1 1 , . . . , n n } in weakly increasing order. Now given an arrangement .ρ = ρ1 . . . ρm , we let .drop(ρ), the number of drops of .ρ, be the number of occurrences where .τi > ρi for .1 ≤ i ≤ m.

Passing Drops and Descents Fig. 1 An example of the arrangement 1324313 of (2) (1) (3) (1) .{1 , 2 , 3 , 4 } with two drops

37 1

2

3

4

1 2 3 4 5 6 7

An alternative way to view drops is in terms of placing rooks on a board subject to certain constraints. In particular, the arrangements of .{1(a1 ) , . . . , n(an ) } are in oneto-one correspondence with rook placements on a board of size .(a1 + · · · + an )×n with one rook in each row and .ai rooks in column i for .1 ≤ i ≤ n. By reading down the rows and recording the column containing the rook for that row we have the arrangement. The number of drops can be found by finding the number of rooks lying in the region below the placement corresponding to .τ (one can think of .τ as the “diagonal”, and so we are looking for rooks below the diagonal). An example of this is shown in Fig. 1 where the region below .τ is shaded. While for any single arrangement the number of drops does not have to equal the number of descents, we will show, as with permutations, that collectively there is a relation. Proposition 1 The number of arrangements of .{1(a1 ) , . . . , n(an ) } with .𝓁 descents is the same as the number of arrangements of .{1(a1 ) , . . . , n(an ) } with .𝓁 drops. Proof Let .S(a1 ,...,an ) be all arrangements of .{1(a1 ) , . . . , n(an ) }. We first recall the well-known case of permutations, .S(1,...,1) (see [3]). Starting with a permutation .π , write it in cycle form, including singleton cycles, where each cycle has its largest element first and cycles are arranged in increasing order with regard to their first element. Reading off the terms in order (e.g. deleting the parentheses from cycle notation) yields a new permutation .^ π. π is bijective since inserting left parentheses before every This map sending .π to .^ maximum as seen from left to right and then inserting the matching right parentheses will uniquely give .π from .^ π . To see that drops of .π map to descents of .^ π , note that π must lie inside a cycle of .π since the ordering of cycles guarantees descents of .^ that the last element of a cycle is always followed by a strictly larger element. Since elements within a cycle are mapped to the next element in the cycle, then descents of .^ π correspond to drops of .π . The key observation we will need for the generalization is that the elements involved in the drops of .π are the same elements involved in the descents of .^ π. We now turn to the general case and consider elements .ρ ∈ S(a1 ,...,an ) . First, we extend .ρ to a collection of permutations of lexicographically ordered elements by replacing the .ai occurrences of i by .(i, 1), (i, 2), . . . , (i, ai ) in all possible ways (this can also be thought of as forming all possible linear extensions of a partial order). There are .a1 ! · · · an ! different extensions. As an example, here is one of the

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all linear extensions

Fig. 2 Intuition behind establishing equal counts for drops and descents

drops

descents

36 ways to extend 121221, 121221 −→ (1, 3)(2, 3)(1, 2)(2, 1)(2, 2)(1, 1).

.

We will split drops of these new permutations into two types: inherited drops from the arrangement where the first elements disagree and new drops where the first elements agree while the second elements disagree. Similarly, we can define inherited descents and new descents. We can now apply the bijection for permutations to the space of all of the linear extensions. By our key observation we have that the bijection not only takes drops to descents, but it takes inherited drops to inherited descents and | new drops to new | | permutations with = 𝓁} descents. In particular, there are .(a| 1 ! . . . an !)|{ρ : drop(ρ) | .𝓁 inherited drops and .(a1 ! . . . an !)|{ρ : des(ρ) = 𝓁}| permutations with .𝓁 inherited descents. By |the bijection the sizes | of these are equal and so we conclude .|{ρ : drop(ρ) = 𝓁}| = |{ρ : des(ρ) = 𝓁}|, as desired. ⨆ ⨅ The basic idea of the preceding proof is shown in Fig. 2 where we take our space of all possible orderings grouped by drops or grouped by descents and then lift up to the space of all linear extensions. Using the bijection for permutations shows that the resulting lifts are the same size, and since the lift scaled both sets by the same amount the original sets are also of the same size. We note that this proof does not give a general bijection, but alternative proofs exist which have explicit bijections, see Han [12]. We denote the〉 number of arrangements of .{1(a1 ) , . . . , n(an ) } with〈 .𝓁〉 descents (or 〈a1 ,...,a drops) by . 𝓁 n ; when .a1 = · · · = an = k we will denote this by . n𝓁 k . / \ / \ a1 , . . . , an aσ (1) , . . . , aσ (n) = Proposition 2 For any permutation .σ , we have . . 𝓁 𝓁 Proof Starting with an arrangement of .{1(a1 ) , . . . , i (ai ) , i + 1(ai+1 ) , . . . , n(an ) }, find all contiguous blocks with elements in .{i, i + 1}. Within each such block, swap i with .i + 1 and then reverse the order of the block. This gives an arrangement of .{1(a1 ) , . . . , i (ai+1 ) , i + 1(ai ) , . . . , n(an ) } with the same number of descents as the original arrangement. To check that the number of descents are the same, note that if there was a descent not involving i or .i + 1, this descent is preserved. Any descent within a contiguous block becomes an ascent when i and .i + 1 are swapped, but becomes a descent

Passing Drops and Descents

39

again when the order is reversed. Additionally, descents involving the first element of a block are preserved because the preceding element before the block must be larger than both i and .i + 1. Similarly, descents involving the last element of a block are preserved. Since this transformation is reversible, this establishes the bijection for any transposition of two consecutive terms. Since any permutation can be formed from a sequence of transposition operations the general case follows. ⨆ ⨅ With this in place we can assume the values in .(a1 , . . . , an ) are arranged in weakly decreasing order. So in particular, one can think about descent/drop statistics being associated with partitions. In Table 1 we have produced the data for some small partitions. Examining the data in Table 1 we see in all cases when the associated partitions have equal-sized parts that the data is symmetric in that row. For the case of permutations this is well-known for descents by the simple argument “write the string in reverse”. For the more general setting this reversal argument does not work. As an example, both 1212 and 2211 have one descent but their reversals 2121 and 〈 〉 n Table 1 A table containing the values of . a1 ,...,a for partitions up through size six 𝓁 .(a1 , . . . , an )

.𝓁

.(1, 1)

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

.(2, 1) .(1, 1, 1) .(3, 1) .(2, 2) .(2, 1, 1) .(1, 1, 1, 1) .(4, 1) .(3, 2) .(3, 1, 1) .(2, 2, 1) .(2, 1, 1, 1) .(1, 1, 1, 1, 1) .(5, 1) .(4, 2) .(4, 1, 1) .(3, 3) .(3, 2, 1) .(3, 1, 1, 1) .(2, 2, 2) .(2, 2, 1, 1) .(2, 1, 1, 1, 1) .(1, 1, 1, 1, 1, 1)

=0

=1 1 2 4 3 4 7 11 4 6 10 12 18 26 5 8 13 9 17 25 20 29 41 57

.𝓁

.𝓁

=2

.𝓁

=3

.𝓁

=4

.𝓁

1 1 4 11

1

3 9 15 33 66

2 8 26

1

6 16 9 33 67 48 93 171 302

1 9 27 20 53 131 302

1 4 16 57

1

=5

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C. Bass and S. Butler

Fig. 3 Establishing symmetry of generalized 〈 〉 Eulerian numbers . n𝓁 k

(a)

(b)

(c)

1122 have two and zero descents respectively. Nevertheless, the symmetry does hold and we will establish it bijectively by use of drops. / \ / \ n n Proposition 3 We have . = . 𝓁k k(n − 1) − 𝓁 k Proof We consider a .(kn)×n board where we place rooks 〈 〉 so that there is one rook in each row and k rooks in each column. Recall that . n𝓁 k is counting the number of rook placements with .𝓁 rooks below the main diagonal. There are kn rooks in total with .𝓁 rooks below the diagonal, k rooks in the last column and so .k(n − 1) − 𝓁 rooks in the remaining part of the board (see Fig. 3a). Now we do two operations, first flip the first .n−1 columns from left-to-right (see Fig. 3b) and then flip the entire board from top-to-bottom (see Fig. 3c). This will put the region with .k(n − 1) − 𝓁 below the diagonal. Moreover, the total number of rooks in each row/column remains the same during this process, and this process is reversible. This gives a bijection between rook placements with .𝓁 drops and rook placements with .k(n − 1) − 𝓁 drops, establishing the result. ⨆ ⨅ An explicit formula for these coefficients can be derived by modifying proofs for Eulerian coefficients (e.g. [2]). We will not need this formula for our purposes, but include it here for completeness. Proposition 4 We have / .

⎛ ⎞ \ ⎲ ⎞∏ 𝓁 n ⎛ a1 , . . . , an ai + 𝓁 − j j a1 + · · · + an + 1 . = (−1) 𝓁 j ai j =0

i=1

Proof The intuitive way to make an arrangement of .{1(a1 ) , . . . , n(an ) } with .𝓁 descents is to set up .𝓁 + 1 buckets in a row and distribute the elements among the buckets. Arrange the elements in each bucket in weakly increasing order and concatenate the results. A problem is some buckets might be empty, or two consecutive buckets might not have a descent since the elements in the first bucket are weakly below all elements in the second. So we correct this with inclusionexclusion.

Passing Drops and Descents

41

Let .N = a1 + · · · + an . Given an arrangement of .{1(a1 ) , . . . , n(an ) } we will look at .N + 1 labeled slots, .{0, 1, . . . , N}, which represent the spaces we can place bars (e.g. between elements as well as the start/end; here bars indicate divisions between buckets). We now consider all arrangements of the elements and bars where between any bars the elements are weakly increasing (multiple consecutive bars are allowed). Let .Pi indicate the subset of this collection where there is a bar in the ith slot that can be removed (e.g. an extra bar in a slot, or the elements occuring on either side of the bar are not a descent). Then we need to find .|P0 (∩ P1 )∩ · · · ∩ PN |. Let .S = {r1 , . . . , rj } ⊆ {0, . . . , N } (there are . N j+1 such sets). Then the elements of .Pr1 ∪ · · · ∪ Prj are in one-to-one correspondence with arrangements of (a ) (a ) .{1 1 , . . . , n n } together with .𝓁 − j bars where between any two bars the elements are weakly increasing (the bijection is to simply pull out/add in a bar out of each gap .r1 , . . . , rj ). In the latter case this can be counted by a bars-and-stars computation for each value. When placing the ( i’s we) have .ai (stars) to place into .𝓁 − j + 1 buckets. For each value i there are . ai +𝓁−j ways to distribute them into the buckets; and ai ∏n (ai +𝓁−j ) . Finally, we now apply inclusionso the ways to place all values is . i=1 ai exclusion, summing up to .𝓁 (there are not more than .𝓁 bars available for removal), to establish the result. ⨆ ⨅

3 Describing Passing Patterns with Cards Passing involves multiple (distinct) jugglers who are throwing objects (usually balls or clubs) amongst themselves. For simplicity we will assume that the juggling happens to a regular beat where in each beat each of the jugglers will catch and then immediately throw an object. Moreover we will only be interested in patterns which are periodic (repeating). We can visualize this process with a passing diagram where we follow the balls over time as they move from beat to beat. We will represent time as a line moving forward from left to right. Because we treat the jugglers as distinct we can consistently order them in each beat; however we emphasize that all jugglers in a given beat are simultaneously catching and throwing and that this convention is for ease of visualization and analysis. An example of a passing diagram is shown in Fig. 4 where we have two jugglers marked as and ; the dotted lines are used to indicate separation between beats. One way to describe the passing pattern is to look at the transitions between what the balls are doing before and after each beat. In particular, we want to look at the relative landing order of the balls if we were to stop juggling (here we take advantage of the ordering of the jugglers if multiple balls land in the same beat to guarantee a unique ordering). An example is shown in Fig. 5 where we stop the passing pattern found in Fig. 4 between two consecutive beats and then mark how the balls land. If the balls were labeled as in the left of Fig. 5 abcdef (ordered by their landing) then in the next beat their relative landing becomes bcdeaf . In particular, we see

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C. Bass and S. Butler

Fig. 4 An example of a passing diagram

ab

c

d e

f

ab

c b c

d e d e

a

f f

Fig. 5 Stopping the passing pattern at two consecutive points in time

f e d c b a

f a e d c b

(a )

(b)

Fig. 6 An example of a card representing one beat (a) and the result of placing cards for each beat (b) for the pattern from Fig. 4

that every beat can be thought of as a permutation of how the balls will land. Each such permutation can be represented by a card where the bottom k balls (where k is the number of jugglers) drop down to the jugglers and then get rearranged by being properly thrown. The remaining .b − k balls represent balls which are not thrown in that beat and for these their relative ordering remains unchanged (e.g. the only action that can change relative ordering is a throw by the jugglers). We can represent the result of each beat then by a permutation, which visually we can represent as a card. As an example the card representing what happens in Fig. 5 is shown in Fig. 6a where the balls are scheduled to land going from bottom-to-top. We can of course repeat this for every beat and then place the cards consecutively. The result of doing this for Fig. 4 is shown in Fig. 6b. If we compare Fig. 4 with Fig. 6b we see that they both are the same passing diagram (one more “stylized”). In general, if we have a passing pattern, then there will be a unique sequence of cards (=permutations at each beat) that represents the pattern. And in the other direction, if we have a sequence of cards, then there is a

Passing Drops and Descents

Fig. 7 All .

43

4! = 12 possible cards for .k = 2 jugglers and .b = 4 balls (4 − 2)!

unique passing pattern for which it corresponds (coming from the passing diagram the cards generate). So we have the following. Proposition 5 There is a bijection between passing patterns with period n among k jugglers with at most b balls where every juggler catches a ball in each beat and sequences of n cards where each card represents a permissible transition given there are k jugglers and b balls. The reason that we have “at most” is that it is possible that the particular set of cards we choose might always have the top few balls never dropping down into the pattern and so the result is a passing pattern not using all of the balls (in practical terms, we might say these are filled with helium). An example of the set of all possible cards for .k = 2 jugglers and .b = 4 balls is shown in Fig. 7. b! different cards which represent a permissible (b − k)! transition given there are k jugglers and b balls where every juggler catches a ball in each beat.

Proposition 6 There are

.

Proof We count the number of permutations that can be generated. This is done by ordering the jugglers and each juggler gets to decide where the unique ball that they receive goes in the new pattern. The first juggler gets b choices, the second get .b −1, and so on to the kth juggler who gets .b − (k − 1) choices. The remaining balls (the balls still in the air) then slot into the remaining positions while still maintaining their relative ordering. This gives the desired count. ⨆ ⨅ Combining the previous two propositions we now have the following result (note that cards can be chosen independently and repetitions are allowed). ⎛ ⎞n b! Proposition 7 There are . different passing patterns of period n among (b − k)! k jugglers with at most b balls where every juggler catches a ball in each beat.

4 Describing Passing Patterns with Rook Placements Another way to describe passing patterns is to focus on what to do with a ball that is currently in the hand and needs to be thrown. This is done by associating each action with a value that indicates how many beats in the future it lands and

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6 4

4 1

1 2

6 4

4 1

1 2

6 4

4 1

1 2

6 4

4 1

Fig. 8 Figure 4 with siteswap indicated under each throw Fig. 9 An overhead cyclic view of the pattern in Fig. 4 to demonstrate Proposition 8. Here the path of the balls can be thought of as string being wrapped around the center clockwise. The total length of the string is the product of the period with the number of balls (nb); and is also equal to the sum total of all the throws

which juggler catches the ball when it lands. Among jugglers, this information on a pattern is known as siteswap and is the “just-in-time” information needed to continue juggling. Given a passing diagram we can find the siteswap by counting the number of beats until it lands and marking the juggler who catches it. As an example, for the pattern shown in Fig. 4 we would have what is shown in Fig. 8. This leads to the siteswap for the two jugglers as follows, : 6 4 1 and : 4 1 2 . For what follows we will need to make use of the following. Proposition 8 The number of balls in a passing pattern of period n can be found by summing all throws among the siteswaps of the jugglers and dividing by n. Proof We count the total amount of time the balls are in the air over one full period in two ways. Since the balls are always automatically caught and thrown then the balls are always in the air and so this value is nb. On the other hand this can be calculated from finding the sum of all the throws; now solving for b gives the result. ⨆ ⨅ Proposition 8 can be visualized by taking a passing pattern and then cutting out one period (e.g. in Fig. 4 this would be three full beats) and then wrapping it around to make a cylinder. We can then “squash” it down and look at it from above to get Fig. 9. This idea of thinking of things cyclically and taking juggling patterns from .Z to .Zn is an important one which we will lean heavily into for what follows. We now recall a key principle in periodic passing. Namely, at every beat each juggler catches a ball and then immediately throws a ball to land at some point in the

Passing Drops and Descents Fig. 10 Boards used to find siteswaps when .k = 2 and .n = 5

Fig. 11 Placing rooks on the board so each row has one rook, and each column collectively among all boards has rooks of all colors

45 ≡1 ≡2 ≡3 ≡4 ≡5 ≡1

≡1 ≡2 ≡3 ≡4 ≡5 ≡1

≡2

≡2

≡3

≡3

≡4

≡4

≡5

≡5 ≡1 ≡2 ≡3 ≡4 ≡5

≡1

≡1 ≡2 ≡3 ≡4 ≡5 ≡1

≡2

≡2

≡3

≡3

≡4

≡4

≡5

≡5

future. Because of the periodic nature of juggling we can think of these operations as happening modulo n. We start by introducing k copies of the .n×n board, one copy for each juggler. We then associate the rows with the timing of the throws and the columns with the timing of the catches (times done modulo n). An example of empty boards with .k = 2 and .n = 5 is in Fig. 10. We now place (colored) rooks on the board to obey the following rules: – There are k different colors, each color is associated with one of the jugglers. – Each row of each board has exactly one rook with no limitation on the coloring of the rook. (This indicates that at each beat each juggler makes one throw to land at some point with some juggler.) – Each column collectively among all the boards has k rooks, one of each color. (This indicates that at each beat each juggler will catch a ball.) An example of a legal placement for rooks for the boards in Fig. 10 is shown in Fig. 11. If we know the row and the column corresponding with a particular throw, then we can determine, modulo n, the throw that a juggler makes at any beat and using the color of the rook determine who catches the ball that will be thrown. If a ball is in the row .≡ i (mod n) and the column .≡ j (mod n), then the throws that can be made are .≡ j − i (mod n). There are many possible passing patterns that can be associated with a placement of rooks as adding n to any throw still satisfies all of the requirements. We will be interested in minimal passing patterns which are the patterns with the minimal throws satisfying the modular conditions (throws only involving numbers from .{1, . . . , n}). In particular, the minimal throw associated with a rook in position row i and column j on a board is ⎧ .

j −i n+j −i

if the rook is strictly above the diagonal, and if the rook is on or below the diagonal.

(1)

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As an example, the placement of the rooks in Fig. 11 corresponds with the minimal siteswap pattern as follows, : 3 2 5 2 3 and : 4 4 4 3 5 . Using Proposition 8, this is a 7-ball pattern. Observation 1 Every valid passing pattern is associated with a unique minimal passing pattern by, if needed, repeatedly subtracting n until all throws are in .{1, . . . , n}. Proposition 9 Given a valid placement of rooks, the number of balls needed for the corresponding minimal passing pattern is the number of rooks on or below the diagonal collectively among all boards. Proof When we add up all of the throws collectively using (1) we see that the contributions from the columns will be .k(1 + 2 + · · · + n) and the contributions from the rows will be .−k(1 + 2 + · · · + n) which cancel. Finally, we will get an n for every rook, collectively, on or below the diagonal. Therefore the sum of the throws is n times the collective number of rooks on or below the diagonal. Now applying Proposition 8 and dividing by n we get the result. ⨆ ⨅ Applying Proposition 9 to Fig. 11, we see that between the two boards there are collectively 7 rooks on or below the main diagonals. Therefore the associated minimal passing pattern would be a 7-ball pattern. The last key ingredient before doing the count for the number of patterns by generating all valid siteswaps is to generate these minimal rook placements. And more particularly we will also need to get a count for the number of rook placements with a fixed number of rooks on or below the diagonal. To do this we will “interlace” the k copies of the .n×n board to produce one single .(kn)×n board. An example of this is shown in Fig. 12. 〈 〉 Proposition 10 There are . n𝓁 k (k!)n minimal passing patterns with k jugglers, period n, and .kn − 𝓁 balls. Proof Start by placing uncolored rooks on a .(kn)×n board with one rook in each row, k rooks in each column, and .kn − 𝓁 rooks on or below the diagonal. This is the

Fig. 12 Interlacing the two .5×5 boards from Fig. 11 to make a .10×5 board with rooks

Passing Drops and Descents

47

same as placing .𝓁 rooks strictly above the diagonal (since there are kn rooks total), since we can rotate the board a half-revolution and still satisfy the counts for the 〈 〉 rows/columns. In particular, there are . n𝓁 k such ways to place rooks. Now for such a placement we color the rooks in each column so that each of the k colors corresponding with the jugglers is present, this can be done in .(k!)n ways. Finally, we note that for each such placement of rooks and coloring by uninterlacing (so that the i-th board has those rows which are .≡ i (mod k) in the same relative ordering) we get a unique minimal passing pattern with k jugglers, period n, and .kn − 𝓁 balls; and every such minimal passing pattern will be formed in this way. 〈 〉 Therefore combining we have . n𝓁 k (k!)n different possible minimal passing patterns. ⨆ ⨅ Proposition 11 The number of siteswaps for passing patterns with k jugglers, period n, and at most b balls where every juggler makes a throw at each beat is ⎲ /n\ .

𝓁

𝓁

⎛ (k!)

n

k

⎞ b+𝓁 . kn

Proof For a given value of .𝓁 we apply Proposition 10 to count the minimal passing patterns with .kn − 𝓁 balls with k jugglers and period n. From Observation 1, all possible patterns can be obtained by adding some multiples of n to the throws of these minimal passing patterns. Moreover, by Proposition 8 if we add n to any single throw then the resulting pattern will need one more ball (i.e. the average of the throws will go up by one). Therefore we can distribute the remaining .b − (kn − 𝓁) balls available for us to use into .kn + 1 locations (one for each throw where here adding a ball means adding n to the value of the throw, and then an extra location for “unused” balls). Using a standard barsand-stars argument, the number of ways this can be done is ⎛( .

) ⎞ ⎛ ⎞ b − (kn − 𝓁) + (kn + 1) − 1 b+𝓁 = . (kn + 1) − 1 kn ⨆ ⨅

Putting it altogether gives the result. We are now ready to prove our main result by comparing our counts.

Proof of Theorem 1 Propositions 7 and 11 both count the number of passing patterns with k jugglers, period n, and at most b balls where every juggler makes a throw at each beat. (One counting by “cards” and the other counting by “siteswaps”.) Comparing the counts we conclude ⎛ .

b! (b − k)!

⎞n =

⎲ /n\ 𝓁

𝓁

k

(k!)n

⎛ ⎞ b+𝓁 . kn

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Dividing both sides by .(k!)n and simplifying, the result is established for all positive whole values of b. Since both sides are polynomials that agree infinitely often, the result holds for arbitrary values of x. ⨆ ⨅

4.1 Avoiding Throws of 0 In establishing Theorem 1 we focused on passing patterns where at each beat, each juggler will throw and catch a ball. In passing, it is possible to relax that assumption and allow jugglers to not make a catch/throw in some beats. In siteswap notation, which records the throws, this is denoted as a throw of 0. There are two main obstacles to using throws of 0 in the proof we have presented. . The number of cards that must be considered grows significantly since we must also identify the subset of jugglers who will not throw in the beat. In particular, there are now k ⎛ ⎞ ⎲ k .

j =0

b! j (b − k + j )!

different cards; which is not consistent with the result of Theorem 1. . For siteswap the situation is worse, in that 0 throws must always occur to the same person. In other words, a 0 throw from one person to another person has no physical interpretation in passing. However, the way that siteswaps were constructed in the proof did not keep track of any limitations on throws. So this would require a significant amount of additional bookkeeping to find the corresponding count for siteswaps with 0 throws. For the simpler case of juggling (passing with 1 person), the first point only adds 1 new card, and the second point cannot happen. So the juggling proof of Worpitzky’s identity is often done with 0 throws allowed.

References 1. Esther Banaian, Steve Butler, Christopher Cox, Jeffrey Davis, Jacob Landgraf, and Scarlitte Ponce, Counting prime juggling patterns, Graphs and Combinatorics 32 (2016), 1675–1688. 2. Miklós Bóna, Combinatorics of Permutations, CRC Press, 2012, xiv+458 pp. 3. Joe Buhler, David Eisenbud, Ron Graham, Colin Wright, Juggling drops and descents, American Mathematical Monthly 101 (1994), 507–519. 4. Joe Buhler Ron Graham, Juggling patterns, passings, and posets, in Mathematical Adventures for Students and Amateurs, MAA, 2004, 99–116. 5. Steve Butler, Jeongyoon Choi, Kimyung Kim, and Kyuhyeok Seo, Enumerating multiplex juggling patterns, Journal of Integer Sequences 22 (2019), 21 pp.

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6. Steve Butler, Fan Chung, Jay Cummings, Ron Graham, Juggling card sequences, Journal of Combinatorics 8 (2017), 507–539. 7. Steve Butler, Ron Graham, Enumerating (multiplex) juggling sequences, Annals of Combinatorics 13 (2010), 413–424. 8. Fan Chung, Ron Graham, Primitive juggling sequences, American Mathematical Monthly 115 (2008), 185–194. 9. Satyan Devadoss and John Mugno, Juggling braids and links, Mathematical Intelligencer 29 (2007), 15–22. 10. Richard Ehrenborg, Margaret Readdy, Juggling and applications to q-analogues, Discrete Math. 157 (1996), 107–125. 11. John Engbers, Jay Pantone, Christopher Stocker, Colored multipermutations and a combinatorial generalization of Worpitzky’s identity, The Australasian Journal of Combinatorics 78 (2020), 335–347. 12. Han, Guo-Niu, Une transformation fondamentale sur les réarrangements de mots, Advances in Mathematics 105 (1994), 26–41. 13. Adam King, Amanda Laubmeier, Kai Orans, Anant Godbole, Universal and overlap cycles for posets, words, and juggling patterns, Graphs and Combinatorics 32 (2016), 1013–1025. 14. Allen Knutson, Randomly juggling backwards, in Connections in discrete mathematics, Cambridge University Press, 2018, 305–320. 15. Claudio Pita-Ruiz, Generalized Eulerian polynomials and some applications, Integers 18 (2018). Paper No. A17, 42 pp. 16. Claudio Pita-Ruiz, On a generalization of Eulerian numbers, Notes on Number Theory and Discrete Mathematics 24 (2018), 16–42. 17. Gregory Warrington, Juggling probabilities, American Mathematical Monthly 112 (2005), 105–118.

Group Divisible Designs with Three Groups and Block Size 4 Dinesh G. Sarvate, Dinkayehu M. Woldemariam, and Li Zhang

Abstract Group divisible designs are classical combinatorial designs studied for their applications as well as for their own sake. They provide an ample opportunity of developing techniques to study combinatorial design constructions. GDDs are inherently hard to construct, especially when the number of groups is less than the block size and group sizes are different. The subject matter for this chapter is GDDs of block size four with three groups of different sizes. A previous study of the problem addressed the cases when the first group size, say .n1 is 1 or 2, the second group size .n2 = n greater than or equal to .n1 and the third group size is .n + 1. The first part of the present paper tackles again the case of the first group having size one and the third group having size .n + 2. We also obtain several non-existence results when restrictions on block configurations are placed. The second part of the chapter deals with group sizes 3, n .(n ≥ 3) and .n + 1, respectively. We hope that these constructions of specific families will help to develop a more unified approach to construct such GDDs. Keywords Combinatorial designs · Groups · Blocks

1 Introduction One of the most useful and well known combinatorial designs is a balanced incomplete block design.

D. G. Sarvate College of Charleston, Charleston, SC, USA e-mail: [email protected] D. M. Woldemariam Adama Science and Technology University, Adama, Ethiopia L. Zhang (✉) The Citadel, Charleston, SC, USA e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 S. Heuss et al. (eds.), Combinatorics, Graph Theory and Computing, Springer Proceedings in Mathematics & Statistics 462, https://doi.org/10.1007/978-3-031-62166-6_4

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Definition 1 A Balanced Incomplete Block Design BIBD.(v, k, λ) is a pair .(V , B) where V is a v-set of points and .B is a collection of k-subsets of V called blocks .(k < v) such that every pair of distinct elements appear in exactly .λ blocks. We will use the following result of Hanani ([4], pp 361–386). Theorem 1 The necessary conditions are sufficient for the existence of a BIBD (v, 4, λ) i.e., a BIBD.(v, 4, λ) exists if and only if .λ ≡ 1, 5 (mod 6) and .v ≡ 1, 4 (mod 12); .λ ≡ 2, 4 (mod 6) and .v ≡ 1 (mod 3); .λ ≡ 3 (mod 6) and .v ≡ 0, 1 (mod 4); .λ ≡ 0 (mod 6) and any v. .

Definition 2 A group divisible design GDD.(n1 , n2 , . . . , nm , k; λ1 , λ2 ) is a triple (X, G, B), where X is a v-set (.v = n1 + n2 + . . . + nm ), .G a partition of X into m subsets (called groups) of size .n1 , n2 , . . . , nm , respectively, and .B is collection of k-subsets of X (called blocks) such that

.

(i) pairs of points within the same group are called first associate of each other and appear together in .λ1 blocks, and (ii) pairs of points not in the same group are called second associates of each other and appear together in .λ2 blocks. When all m groups are of the same size, say n, then a GDD.(n, n, . . . , n, k; λ1 , λ2 ) is denoted by GDD.(n, m, k; λ1 , λ2 ). Group divisible designs (GDDs) have been studied for their usefulness in statistics and for their universal application to constructions of new designs [11, 19] and [20] including BIBDs. In [2] and [3], the question of existence of GDDs for block size three was settled. There is a more technical proof given in the book “Triple Systems” [1]. Similar results were established for GDDs with block size four in [5, 7, 9, 15] and [21]. Also, results about GDDs with two groups and block size four with equal number of even and odd blocks were given in [14]. In [17], results on GDDs with number of groups 2 or 3 and block size four were established. Furthermore, in [6] and [8], results about GDDs with two groups, and block size five with fixed block configuration were presented. In [18], results about GDDs with three groups, and block size five with fixed block configuration were addressed. In [12], results about GDDs with fours groups, and block size five with fixed block configuration were established. In [16], generalizations of designs for block size five from Clatworthy’s Table are given. In [10], results about GDDs with block size six with fixed block configuration were studied. Certain difficulties are present in the constructions of GDDs especially when the number of groups is smaller than the block size. Furthermore, if the sizes are different, the cases and subcases to deal with the existence problem increase and make it more challenging to obtain a complete result. As GDD.(n, 3, 4; λ1 , λ2 ) has 3 groups and the block size is 4, there are 4 types of blocks: blocks of configuration .(0, 4), .(1, 3), .(2, 2) and .(1, 1, 2) depending on the number of elements from distinct groups. To be clear, a block of configuration

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(a, b) means that the block contains a point from one group and b points from one of the remaining two groups. Similarly, the blocks of configuration .(1, 1, 2) means the block has one point each from two of the three groups and two points from the third group. GDDs with block configurations were studied in [5] and [21] including GDD.(n, 3, 4; λ1 , λ2 )’s with blocks having configuration .(1, 1, 2). When the number of groups is 2 and the block size is 4, there are three types of blocks for a GDD.(n, 2, 4; λ1 , λ2 ): blocks of configuration .(0, 4), .(1, 3) and .(2, 2). In [14] results on GDD.(n, 2, 4; λ1 , λ2 )’s with equal number of blocks having configuration .(1, 3) and .(2, 2) are given. Note that a GDD.(n1 , n2 , n3 , k; λ, λ) is just a BIBD.(n1 + n2 + n3 , k, λ). Also, a GDD.(n1 , n2 , n3 , 4; λ1 , λ2 ) exists when .λ2 ≤ λ1 and .λ1 ≡ λ2 (mod 6), if a BIBD.(n1 + n2 + n3 , 4, λ2 ) exists as a BIBD.(v, 4, 6) exists for all integers .v ≥ 4. More generally, .

Theorem 2 If a BIBD.(n1 + n2 + n3 , 4, λ2 ) and a BIBD.(ni , 4, λ1 ) exist for .i = 1, 2, 3, then a GDD.(n1 , n2 , n3 , 4; λ1 + λ2 , λ2 ) exists. Corollary 1 A GDD.(n1 , n2 , n3 , 4; λ1 = λ2 + 6s, λ2 ) exists if a BIBD .(n1 + n2 + n3 , 4, λ2 ) exists and .ni ≥ 4 for .i = 1, 2, 3. Corollary 2 For all values of .n1 , n2 , n3 where .n1 + n2 + n3 ≡ 1, 4 (mod 12) and a BIBD.(ni , 4, λ) exists for .i = 1, 2, 3, a GDD.(n1 , n2 , n3 , 4; λ1 .= λ + λ2 , λ2 ) exists for all .λ2 ’s. In [5] and [21], the necessary conditions are proved to be sufficient for the existence of a GDD.(n, 3, 4; λ1 , λ2 ) with configuration .(1, 1, 2), i.e. each block in a design contains an element from the two of the three groups, respectively, and the two remaining elements in the block are from the third group. Note that three groups have the same size n in this problem. In [13], some general results were established for GDD.(n1 , n, n + 1, 4; λ1 , .λ2 ) where .n1 = 1 or .n1 = 2. In the first part of this paper, we study the existence problem of GDD.(1, n, n + 2, 4; λ1 , .λ2 ). We also find non-existence results with configuration restriction for GDD.(1, n, n + t, 4; λ1 , .λ2 ) for any positive integer t which are given in the Appendix. In the second part of the paper we study GDD.(3, n, n + 1, 4; λ1 , λ2 ).

2 The Necessary Conditions for GDD(1, n, n + 2, 4; λ1 , λ2 ) Let b represent the number of blocks and .ri represent the number of appearances (called the replication number) of each point from group .Gi in a GDD. For .n = 1, a GDD.(1, 1, 3, 4; λ1 , λ2 ) exists only when .λ1 = λ2 and it is a BIBD.(5, 4, λ1 ) on .G1 ∪ G2 ∪ G3 . As BIBD.(5, 4, 3) exists, we have the following. Lemma 1 A GDD.(1, 1, 3, 4; λ1 , λ2 ) exists only when .λ1 = λ2 and its blocks are λ1 copies of BIBD.(5, 4, 3) on .G1 ∪ G2 ∪ G3 .

.

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Proof BIBD.(5, 4, 3) on .V = G1 ∪ G2 ∪ G3 can be obtained by taking all the 4subsets of V . Hence, taking .λ1 copies of all the 4-subsets of V provide the required ⨆ ⨅ GDD. Example 1 A GDD.(1, 1, 3, 4; 3, 3) with .G1 = {x}, G2 = {a} and .G3 = {1, 2, 3}. Here .b = 5, .r1 = r2 = r3 = 4. The blocks of the GDD are given below in columns. x a . 1 2

x a 2 3

x a 3 1

x 1 2 3

a 1 2 3

As a result of the above lemma, we can assume that .n > 1 throughout this section and Sect. 3. In this section we will find necessary conditions for the existence of a GDD.(1, n, .n + 2, 4; λ1 , λ2 ). Assuming a GDD.(1, n, n + 2, 4; λ1 , λ2 ) exists, by a counting argument, the replication numbers .ri for the elements of .i th group are: .r1 = (2n+2)λ2 2 2 , .r2 = (n−1)λ1 +(n+3)λ and .r3 = (n+1)λ1 +(n+1)λ . 3 3 3 +4n+2)λ2 Since .4b = r1 + nr2 + (n + 2)r3 , .b = (n +n+1)λ1 +(n . 6 Using the fact that the parameters .r1 , r2 , r3 and b must be integers, we obtain certain restrictions on n which are given in Table 1 where .λ1 and .λ2 are both expressed in terms of congruence modulo 6, and “None” means the design does not exists for any n. 2

2

Lemma 2 ([13]) A necessary condition for the existence of a GDD.(n1 , .n2 , n3 , 4; λ1 , λ2 ) is .b ≥ max.(2ri − λ1 ), .i = 1, 2, 3. Note that when .n1 = 1, .b ≥ max.(2ri − λ1 ), .i = 2, 3.

.

An example that does not satisfy the above necessary condition is given below. Example 2 A GDD.(1, 2, 4, 4; 6, 9) does not exist as .b = 28, .r1 = 18, r2 = 17, r3 = 15 and .28 = b < 2r1 − λ1 = 30.

.

The following is an example of a GDD that exists and satisfies the above necessary conditions.

Table 1 The necessary conditions for GDD.(1, n, n + 2, 4; λ1 , λ2 ) .λ1 \λ2

0 1 2 3 4 5

0 all n None None None None None

1 2 3 None None n even .n ≡ 5 (mod 6) None None None .n ≡ 2, 5 (mod 6) None None None n odd .n ≡ 2 (mod 6) None None None None None

4 5 None None None None None .n ≡ 2 (mod 6) None None .n ≡ 2, 5 (mod 6) None None .n ≡ 5 (mod 6)

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Example 3 A GDD.(1, 2, 4, 4; 6, 3) with .G1 = {x}, G2 = {a, b} and .G3 = {1, 2, 3, .4}. Here .b = 14, .r1 = 6, .r2 = 7 and .r3 = 9. The blocks of the GDD are given below in columns. x a . b 1

x a b 2

x a 3 4

x b 3 4

x 1 2 4

x 1 2 3

a b 1 4

a b 2 3

a b 3 1

a b 4 2

1 2 3 4

11 22 33 44

1 2 3 4

Observe that every block of a GDD.(1, n, n + 2, 4; λ1 , λ2 ) contains at least two first associate pairs except for the blocks that contain the element of .G1 . Then the .r1 blocks that contain the element of .G1 contain at least 1 first associate pair. So we must have at least .r1 + 2(b − r1 ) = 2b − r1 first associate pairs. This implies .[n2 + n + 22]λ1 ≥ 2b − r1 . Hence, we have Lemma 3 A necessary condition for the existence of a GDD.(1, n, n + 2, 4; λ1 , λ2 ) is .λ1 (n2 + n + 1) ≥ 2b − r1 . The following example does not satisfy the above necessary condition. Example 4 A GDD.(1, 2, 4, 4; 2, 5) does not exist as .r1 = 10, r2 = 9, r3 = 7 and b = 14. But .14 = (n2 +n+1)λ1 ≱ 2b−r1 = 26. Also a GDD.(1, 8, 10, 4; 2, 5) does not exist as .r1 = 30, r2 = 23, r3 = 21 and .b = 106. But .146 = (n2 + n + 1)λ1 ≱ 2b − r1 = 182.

.

Though general results are hard to obtain, we have an interesting bound for how far away .λ2 can go from .λ1 . Theorem 3 A necessary condition for the existence of a GDD.(1, n, n + 2, 4; λ1 , λ2 ) is .λ2 < 2λ1 .

.

Proof Substituting the values of b and .r1 in .λ1 (n2 + n + 1) ≥ 2b − r1 , we have 2(n2 +n+1)λ1 .λ2 ≤ < 2λ1 for .n ≥ 2. ⨆ ⨅ n2 +2n

3 General Construction of GDD(1, n, n + 2, 4; λ1 , λ2 ) In this section we will study the existence of a GDD.(1, n, n + 2, 4; λ1 , λ2 ) where G1 , .G2 and .G3 be groups with size 1, n and .n + 2, respectively. To do this we will use BIBDs with block size 4 through out this section. A GDD.(1, n, n + 2, 4; λ1 , λ2 ) exists when .λ2 ≤ λ1 and .λ1 ≡ λ2 (mod 6) if a BIBD.(2n + 3, 4, λ2 ) exists as a BIBD.(n, 4, 6) exists for .n ≥ 4. More generally, we have the following theorem:

.

Theorem 4 If a BIBD.(2n+3, 4, λ2 ), a BIBD.(n, 4, λ) and a BIBD.(n+2, 4, λ) exist, then a GDD.(1, n, n + 2, 4; λ1 = λ + λ2 , λ2 ) exists.

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Proof Assume that a BIBD.(2n+3, 4, λ2 ), a BIBD.(n, 4, λ) and a BIBD.(n+2, 4, λ) exist. Then the blocks of a BIBD.(2n + 3, 4, λ2 ) on .G1 ∪ G2 ∪ G3 along with the blocks of a BIBD.(n, 4, λ) on .G2 and the blocks of a BIBD.(n + 2, 4, λ) on .G3 provide the blocks of a GDD.(1, n, n + 2, 4; λ1 = λ + λ2 , λ2 ). ⨆ ⨅ For example, a GDD.(1, 8, 10, 4, 12, 6) exists by Theorem 4. Corollary 3 A GDD.(1, n, n + 2, 4; λ1 = 6s + λ2 , λ2 ) exists for .n ≡ 5 (mod 6) and any non-negative integer s. Proof For .n ≡ 5 (mod 6), a BIBD.(n, 4, 6) and a BIBD.(n+2, 4, 6) exist. As .n ≡ 5 (mod 6) implies .2n + 3 ≡ 1 (mod 12), a BIBD.(2n + 3, 4, 1) exists. Then taking the blocks of .λ2 copies of a BIBD.(2n + 3, 4, 1) on .G1 ∪ G2 ∪ G3 along with the blocks of s copies of a BIBD.(n, 4, 6) on .G2 and the blocks of s copies of a BIBD(n + 2, 4, 6) on .G3 provide the required GDD. ⨆ ⨅ Corollary 4 A GDD.(1, n, n + 2, 4; λ1 = 6s + λ2 , λ2 ) exists for .n ≡ 2 (mod 6), λ2 ≡ 0 (mod 2), .n /= 2 and any non-negative integer s.

.

Proof Let .λ2 = 2t for some positive integer t. For .n ≡ 2 (mod 6), a BIBD.(n, 4, 6) and a BIBD.(n + 2, 4, 6) exist. As .n ≡ 2 (mod 6) implies .2n + 3 ≡ 1 (mod 3), a BIBD.(2n + 3, 4, 2) exists. Then the blocks of t copies of a BIBD.(2n + 3, 4, 2) on .G1 ∪ G2 ∪ G3 along with the blocks of s copies of a BIBD.(n, 4, 6) on .G2 and the blocks of s copies a BIBD.(n + 2, 4, 6) on .G3 provide the required GDD. ⨆ ⨅ .

Corollary 5 A GDD.(1, n, n + 2, 4; λ1 = 6s + λ2 , λ2 ) exists for .n ≡ 3 (mod 6), λ2 ≡ 0 (mod 3), .n /= 3 and any non-negative integer s.

.

Proof Let .λ2 = 3t for some positive integer t. For .n ≡ 3 (mod 6), a BIBD.(n, 4, 6) and a BIBD.(n + 2, 4, 6) exist. As .n ≡ 3 (mod 6) implies .2n + 3 ≡ 1 (mod 4), a BIBD.(2n + 3, 4, 3) exists. Then the blocks of t copies of a BIBD.(2n + 3, 4, 2) on .G1 ∪ G2 ∪ G3 along with the blocks of s copies of a BIBD.(n, 4, 6) on .G2 and the blocks of s copies a BIBD.(n + 2, 4, 6) on .G3 provide the required GDD. ⨆ ⨅ .

Theorem 5 If a GDD.(n1 , n2 , k; λ1 , λ2 ) and a BIBD.(n1 + n2 , k − 1, λ) with .r = λ2 + λ exist, then a GDD.(1, n1 , n2 , k; λ1 + λ, λ2 + λ) exists. Proof Assume that a GDD.(n1 , n2 , k; λ1 , λ2 ) and a BIBD.(n1 +n2 , k−1, λ) with .r = λ2 +λ exist on .G2 ∪G3 . Then taking the union of each block of a BIBD.(n1 +n2 , k − 1, λ) with the point of .G1 gives blocks of size k in which each first associate pair occurs .λ times, each second associate pair containing the point of .G1 occurs .λ2 + λ times and each second associate pair containing elements from .G2 and .G3 occurs .λ times. So the above blocks together with the blocks of GDD.(n1 , n2 , k; λ1 , λ2 ) contain .λ1 + λ first associate pairs and .λ2 + λ second associate pairs. Hence, a GDD.(1, n1 , n2 , k; λ1 + λ, λ2 + λ) can be obtained by taking the union of blocks of a GDD.(n1 , n2 , k; λ1 , λ2 ) on .G2 ∪ G3 along with the blocks obtained by adjoining each block of BIBD.(n1 + n2 , k − 1, λ) with the point of .G1 . ⨆ ⨅

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Example 5 A GDD.(1, 2, 4, 4; 8, 5) with groups .G1 , .G2 , .G3 of size 1, 2, 4, respectively, can be constructed by taking union of the blocks of a GDD.(2, 4, 4; .6, 3) on .G2 ∪ G3 along with the blocks obtained by adjoining the point of .G1 with each block of BIBD.(6, 3, 2) on .G2 ∪ G3 . The blocks of GDD.(2, 4, 4; 6, 3) can be obtained by taking the union of .G2 with every 2-subset of .G3 along with 5 copies of .G3 . Notice that we can not use Theorem 5 to construct a GDD.(1, 4, 6, 4; 6, 9) as a BIBD.(10, 3, 2) with replication number 9 exists but a GDD.(4, 6, 4; 4, 7) does not exist.

3.1 GDD(1, n, n + 2, 4; λ1 , λ2 ) When λ1 ≤ 3 We are considering .λ1 ≤ 3 as in these cases GDDs are just BIBDs. We start with λ1 = 3.

.

GDD.(1, n, n + 2, 4; 3, λ2 ) For .λ1 = 3, .λ2 < 6 from Theorem 3. A GDD.(1, n, n + 2, 4; 3, 3) is just a BIBD.(2n + 3, 4, 3) and a BIBD.(2n + 3, 4, 3) exists when .n ≡ 1 (mod 2). Also, from Table 1, a GDD.(1, n, n+2, 4; 3, λ2 ) for .λ2 = 0, 1, 2, 4, 5 does not exist. Hence, we have the following result. Lemma 4 A GDD.(1, n, n + 2, 4; 3, λ2 ) does not exist except when .λ2 = 3 and in this case it exists only for .n ≡ 1 (mod 2) and it is a BIBD.(2n + 3, 4, 3). GDD.(1, n, n+2, 4; 1, λ2 ) For .λ1 = 1, .λ2 ≤ 1 from Theorem 3. Also, from Table 1, GDD.(1, n, n + 2, 4; 1, λ2 ) does not exist for .λ2 = 0. So a GDD.(1, n, n + 2, 4; 1, 1) is just a BIBD.(2n + 3, 4, 1) which exists when .n ≡ 5 (mod 6). Thus we have Lemma 5 A GDD.(1, n, n + 2, 4, 1, λ2 ) does not exist for any .λ2 except when .λ2 = 1 and in this case it exists only for .n ≡ 5 (mod 6) and it is just a BIBD.(2n+3, 4, 1). GDD.(1, n, n + 2, 4; 2, λ2 ) When .λ1 = 2, from Theorem 3 we have .λ2 ≤ 3. Also, from Table 1, one can easily see that a GDD.(1, n, n + 2, 4; 2, λ2 ) does not exist for .λ2 = 0, 1 or 3. A GDD.(1, n, n + 2, 4; 2, 2) is just a BIBD.(2n + 3, 4, 2) which exists when .n ≡ 2 (mod 3). Therefore, we have Lemma 6 A GDD.(1, n, n + 2, 4, 2, λ2 ) does not exist for any .λ2 except when .λ2 = 2 and in this case it exists only for .n ≡ 2 (mod 3) and it is just a BIBD.(2n+3, 4, 2). In summary, a GDD.(1, n, n + 2, 4; λ1 , λ2 ) for .λ1 ≤ 3 exists only when .λ1 = λ2 and in that case it exists if and only if a BIBD.(2n + 3, 4, λ1 ) exists.

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3.2 GDD(1, n, n + 2, 4; λ1 , λ2 ) When λ1 ≡ λ2 (mod 6) In this subsection, we study a GDD.(1, n, n + 2, 4; λ1 , λ2 ) for the cases on the main diagonal of Table 1. Here we consider when .λ1 ≥ λ2 and .λ1 < λ2 . λ1 ≥ λ2

.

λ1 ≡ 0 (mod 6) and .λ2 ≡ 0 (mod 6) For any value of n, a GDD.(1, n, n +2, 4; 6s, 6t) is possible, but it does not exist in some situations. From Theorem 3, a GDD.(1, n, n + 2, 4; 6s, 6t) does not exist for some positive integers .t ≥ 2s. For instance, a GDD.(1, n, n + 2, 4; 6, 12) does exist for any .n ≥ 2 by Lemma 3.

. .

Lemma 7 For all values of .n ≥ 4, a GDD.(1, n, n + 2, 4; 6s, 6t) exists for some positive integers s and t with .t ≤ s. Proof The blocks of GDD.(1, n, n + 2, 4; 6s, 6t) can be obtained by taking union of the collection of blocks of BIBD.(2n+3, 4, 6t) on .G1 ∪G2 ∪G3 , a BIBD.(n, 4, 6(s − t)) on .G2 and BIBD.(n + 2, 4, 6(s − t)) on .G3 . ⨆ ⨅ λ1 ≡ 1 (mod 6) and .λ2 ≡ 1 (mod 6) From Table 1 for this case, .n ≡ 5 (mod 6). Then by Corollary 3, the required GDD exists.

.

λ1 ≡ 2 (mod 6) and .λ2 ≡ 2 (mod 6) From Table 1 for this case, .n ≡ 2, 5 (mod 6). Hence, by Corollary 3 and Corollary 4, we have the required GDD.

.

λ1 ≡ 3 (mod 6) and .λ2 ≡ 3 (mod 6)

.

Lemma 8 Necessary conditions are sufficient for the existence of a GDD.(1, .n, n + 2, 4; λ1 ≡ 3 (mod 6), λ2 ≡ 3 (mod 6)) for .λ2 ≤ λ1 . Proof From Table 1, for this case n is odd, say .n = 2t + 1 for some positive integer t. As .4t + 5 ≡ 1 (mod 4), a BIBD.(4t + 5, 4, λ2 ) exists. Then the blocks 2 of BIBD.(4t + 5, 4, λ2 ) on .G1 ∪ G2 ∪ G3 along with the blocks of . λ1 −λ copies of 6 BIBD.(2t + 1, 4, 6) on .G2 and BIBD.(2t + 3, 4; 6) on .G3 provide the blocks of a GDD.(1, 2t +1, 2t +3, 4; λ1 , λ2 ). That is, a GDD.(1, 2t +1, 2t +3, 4; 6r +3, 6s +3) when .r ≥ s where r and s are nonnegative integers. ⨆ ⨅ λ1 ≡ 4 (mod 6) and .λ2 ≡ 4 (mod 6) From Table 1, for this case .n ≡ 2, 5 (mod 6) and it is identical to the case when .λ1 ≡ 2 (mod 6) and .λ2 ≡ 2 (mod 6). Then a GDD.(1, n, n + 2, 4; λ1 ≡ 4 (mod 6), λ2 ≡ 4 (mod 6)) can be obtained by taking 2 copies of the blocks of a GDD.(1, n, n + 2, 4; λ1 ≡ 2 (mod 6), λ2 ≡ 2 (mod 6)). Hence, we proved the existence of the required GDD.

.

λ1 ≡ 5 (mod 6) and .λ2 ≡ 5 (mod 6) From Table 1, for this case .n ≡ 5 (mod 6). Then by Corollary 3 we have the required GDD. In summary, the above subcases together prove the following theorem.

.

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Theorem 6 Necessary conditions are sufficient for the existence of a GDD.(1, n, n + 2, 4; λ1 , λ2 ) when .λ1 ≥ λ2 and .λ1 ≡ λ2 (mod 6).

.

λ1 < λ2 Let .λ1 ≡ 1 (mod 6), say .n = 6t + 5 for some integer t. Then from Table 1, .λ2 ≡ 1 (mod 6). But .λ2 < 2λ1 by Theorem 3. As .λ1 < λ2 , the only choices for .λ2 are from .{6s + 7, 6s + 13, . . . , 12s + 1} for some integer s. So when .λ1 = 1, GDD.(1, n, n + 2, 4; 1, λ2 ) does not exist. As .n ≡ 5 (mod 6) when .λ1 ≡ 1 (mod 6), the smallest design to construct will be a GDD.(1, 5, 7, 4; 7, 13). But this design does not exist by Lemma 3. For .λ1 = 13, .λ2 < 26 implies that .λ2 = 19 or 25. But GDD.(1, 5, 7, 4; 13, 25) does not exist. Also for .λ1 = 19, .λ2 < 38 implies that .λ2 = 25, 31 or 37 and GDD.(1, 5, 7, 4; 19, 37) does not exist. For .n ≡ 5 (mod 6), when .λ1 = 1 and .λ2 = 6s + 1 for some positive integer s, 2 .λ1 (n + n + 1) ≱ 2b − r1 . Hence, we have the following: .

Lemma 9 A GDD.(1, n, n + 2, 4; 1, 6s + 1) for .n ≡ 5 (mod 6) and a GDD.(1, 5, 7, 4; λ1 = 6s + 1, λ2 = 12s + 1) do not exist for any positive integer s.

.

Since we are considering the cases on the main diagonal entries of Table 1, if we let .λ1 ≡ 2 (mod 6), say .λ1 = 6s + 2 for some integer s, then .λ2 ≡ 2 (mod 6). But for the existence of GDD.(1, n, n + 2, 4; λ1 , λ2 ), .λ2 < 2λ1 by Theorem 3. As .λ1 < λ2 , the only choice for .λ2 is from .{6s + 8, 6s + 14, . . . , 12s + 2}. So when .λ1 = 2, GDD.(1, n, n + 2, 4; 2, λ2 ) does not exist. As .n ≡ 2, 5 (mod 6) when .λ1 ≡ 2 (mod 6), the smallest design to construct is a GDD.(1, 2, 4, 4; 8, 14). But this design does not exist. For .λ1 = 14, .λ2 < 28 implies that .λ2 = 20 or 26 and a GDD.(1, 2, 4, 4; 14, 20), a GDD.(1, 2, 4, 4; 14, 26) and a GDD.(1, 5, 7, 4; 14, 26) do not exist. For .λ1 = 20, .λ2 can be .26, 32, or 38 but a GDD.(1, 2, 4, 4; 20, 26), a GDD.(1, 2, .4, 4; 20, 32), a GDD.(1, 2, 4, 4; 20, 38) and a GDD.(1, 5, 7, 4; 20, 38) do not exist. Moreover, for .n ≡ 2, 5 (mod 6), .λ1 = 2 and .λ2 = 6s + 2 for some positive integer s, .λ1 (n2 + n + 1) ≱ 2b − r1 . So we have the following: Lemma 10 A GDD.(1, n, n + 2, 4; 2, 6s + 2) for .n ≡ 2, 5 (mod 6) and a GDD.(1, 2, 4, 4; 6s + 2, 12s + 2) do not exist for any positive integer s and a GDD.(1, 5, 7, 4; .6s + 2, 12s + 2) does not exist for any positive integer .s > 1. .

By the same argument, it can be easily shown that a GDD.(1, n, n + 2, 4; 3, 6s + 3), a GDD.(1, n, n + 2, 4; 4, 6s + 4) and a GDD.(1, n, n + 2, 4; 5, 6s + 5) for .n ≡ 1, 3, 5 (mod 6), .n ≡ 2, 5 (mod 6) and .n ≡ 5 (mod 6) respectively do not exist for some positive integer s. Moreover, a GDD.(1, 2, 4, 4; 6s + 9, 12s + 15) and a GDD.(1, 2, 4, 4; 6s + 4, 12s + 4) do not exist for a nonnegative integer s as .λ1 (n2 + n + 1) ≱ 2b − r1 .

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3.3 GDD(1, n, n + 2, 4; λ1 , λ2 ) with the Cases on the Off-Diagonal Entries of Table 1 λ1 ≡ 0 (mod 6) and .λ2 ≡ 3 (mod 6) For .n ≥ 4, BIBD.(n, 4, 6) and BIBD.(n + 2, 4, 6) exist and hence a GDD.(1, n, .n + 2, 4; 6t, 0) exists for some positive integer t. But a GDD.(1, n, n + 2, 4; 0, λ2 ) does not exist for any n as the size of the block is greater than the number of groups. Hence, a GDD.(1, n, n + 2, 4; 0, 3) does not exist for .n ≡ 0, 2, 4 (mod 6). Although, it is difficult to obtain general existence result for a GDD.(1, n, n + 2, 4; 6, 3) when .n ≡ 0, 2, 4 (mod 6), we may construct such GDD for small values of n. For instance, a GDD.(1, 2, 4, 4; 6, 3) and a GDD.(1, 2, 4, 4; 8, 5) exist (see Examples 3, 5).

.

λ1 ≡ 2 (mod 6) and λ2 ≡ 5 (mod 6) Consider a GDD(1, n, n+2, 4; 2, 5) for n ≡ 2 (mod 6). Let n = 6t +2 for some nonnegative integer t. Then r1 = 20t +10, r2 = 14t +9, r3 = 14t +7 and b = 42t 2 +50t +14. Since λ1 (n2 +n+1) = 72t 2 +60t +14 and 2b − r1 = 84t 2 + 80t + 18, by Lemma 3, it does not exist. Hence, we have Lemma 11 A GDD(1, n, n + 2, 4, 2s, 5s) does not exist for n ≡ 2 (mod 6) and any positive integer s. λ1 ≡ 4 (mod 6) and λ2 ≡ 1 (mod 6) Suppose a GDD(1, 2, 4, 4; 4, 1) exists. Then r1 = 2, r2 = 3, r3 = 5 and b = 7. As every pair of elements of Gi for i = 2, 3 occurs in exactly 4 blocks, each element of G2 occurs 4 times in the design. But this is a contradiction as r2 = 3. Thus, a GDD(1, 2, 4, 4; 4, 1) does not exist. As a summary of this section, we have the following theorem. Theorem 7 Necessary conditions are sufficient for the existence of a GDD(1, n, n + 2, 4; λ1 , λ2 ) when λ1 ≥ λ2 and λ1 ≡ λ2 (mod 6) except for n even, λ1 ≡ 0 (mod 6) and λ2 ≡ 3 (mod 6), and for n ≡ 2 (mod 6), λ1 ≡ 2, 4 (mod 6) and λ2 ≡ 1, 5 (mod 6).

4 GDD(3, n, n + 1, 4; λ1 , λ2 ) In [5] and [21], the necessary conditions are proved to be sufficient for the existence of a GDD.(n, 3, 4; λ1 , λ2 ) with Configuration .(1, 1, 2), i.e. each block in a design contains an element from the two of the three groups, respectively, and the two remaining elements in the block are from the third group. Note that three groups have the same size n in this problem. In [13], some general results were established for GDD.(n1 , n, n + 1, 4; λ1 , .λ2 ) where .n1 = 1 or .n1 = 2. In the next sections we study GDD.(n1 , n, n + 1, 4; λ1 , .λ2 ) for .n1 = 3 and .n ≥ 3. If .n1 = 3, n2 = n and .n3 = n+1, a GDD.(3, n, n+1, 4; λ, λ) exists whenever a BIBD.(2n+4, 4, λ) exists. We have the following result from Corollary 1.

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Corollary 6 A GDD.(3, n, n + 1, 4; λ, λ) exists if .n ≡ 0 (mod 6). Also, a GDD.(3, n, n + 1, 4; λ, λ) exists if .n ≡ 2, 4 (mod 6) and .λ ≡ 0, 3 (mod 6), and a GDD.(3, n, n + 1, 4; λ, λ) exists if .n ≡ 1, 5 (mod 6) and .λ ≡ 0 (mod 6), and a GDD.(3, n, n + 1, 4; λ, λ) exists if .n ≡ 3 (mod 6) and .λ ≡ 0, 2, 4 (mod 6).

4.1 Necessary Conditions for GDD(3, n, n + 1, 4; λ1 , λ2 ) Suppose a GDD.(3, n, n + 1, 4; λ1 , λ2 ) (exists Notice that the ) ( and ) (has )b blocks. 2 + 3, and the number number of the first associate pairs equals . 32 + n2 + n+1 = n 2 of the second associate pairs equals .3(2n + 1) + n(n + 1) = n2 + 7n + 3. Since each () 2 2 block of size 4 has . 42 = 6 pairs, the number of blocks .b = λ1 (n +3)+λ62 (n +7n+3) . Since b must be an integers, we have the following. – – – – – –

n≡0 n≡1 .n ≡ 2 .n ≡ 3 .n ≡ 4 .n ≡ 5 . .

(mod (mod (mod (mod (mod (mod

6): .λ1 + λ2 ≡ 0 (mod 2), i.e., .λ1 ≡ λ2 (mod 2). 6): .4λ1 + 5λ2 ≡ 0 (mod 6). 6): .λ1 + 3λ2 ≡ 0 (mod 6). 6): .λ2 ≡ 0 (mod 2), i.e., .λ2 is even. 6): .λ1 + 5λ2 ≡ 0 (mod 6). 6): .4λ1 + 3λ2 ≡ 0 (mod 6).

The replication number for an arbitrary point in group .G1 is .r1 = 2λ1 +λ23(2n+1) . 2 (n+4) and .r3 = λ1 n+λ32 (n+3) . Since .r1 , r2 and .r3 must be Similarly, .r2 = λ1 (n−1)+λ 3 integers, we have the following. – .n ≡ 0 (mod 3): From .r1 , we have .2λ1 + λ2 ≡ 0 (mod 3). From .r2 , we have .λ2 − λ1 ≡ 0 (mod 3). Since there is no condition on .r3 , .λ2 − λ1 ≡ 0 (mod 3) for this case. – .n ≡ 1 (mod 3): From .r1 , we have .λ1 ≡ 0 (mod 3). From .r2 , we have .λ2 ≡ 0 (mod 3). From .r3 , we have .λ1 + λ2 ≡ 0 (mod 3). Hence, .λ1 ≡ 0 (mod 3) and .λ2 ≡ 0 (mod 3). – .n ≡ 2 (mod 3): From both .r1 and .r3 , we have .λ1 + λ2 ≡ 0 (mod 3). From .r2 , we have .λ1 ≡ 0 (mod 3). Hence, .λ1 ≡ 0 (mod 3) and .λ2 ≡ 0 (mod 3). Combining the cases above, we have the following. – .n ≡ 0 – .n ≡ 1 – .n ≡ 2 parity. – .n ≡ 3 – .n ≡ 4 parity. – .n ≡ 5

(mod 6): .λ2 − λ1 ≡ 0 (mod 6). (mod 6): .λ1 ≡ 0 (mod 3) and .λ2 ≡ 0 (mod 6). (mod 6): .λ1 ≡ 0 (mod 3), .λ2 ≡ 0 (mod 3), and .λ1 and .λ2 have the same (mod 6): .λ2 − λ1 ≡ 0 (mod 3) and .λ2 is even. (mod 6): .λ1 ≡ 0 (mod 3), .λ2 ≡ 0 (mod 3), and .λ1 and .λ2 have the same (mod 6): .λ1 ≡ 0 (mod 3) and .λ2 ≡ 0 (mod 6).

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Notice that .λ1 should be less than or equal to the minimum of .{r1 , r2 , r3 } in general. Since .r1 = min .{r1 , r2 , r3 } for GDD.(3, n, n + 1, 4; λ1 , λ2 ), .λ1 ≤ r1 = 2λ1 +λ2 (2n+1) , i.e., .λ1 ≤ λ2 (2n + 1). Also, since every block has at least one first 3 () ( ) ( ) λ1 (n2 +3)+λ2 (n2 +7n+3) ≤ associate pair, we have .b ≤ λ1 ( 32 + n2 + n+1 2 ), i.e. . 6

+7n+3) ≤ λ1 ≤ λ1 (n2 + 3). Simplifying this and combining the above, we have . λ2 (n 5(n2 +3) λ2 (2n + 1). From the necessary conditions, if .λ1 is even and .λ2 is odd, a GDD.(3, n, n + 1, 4; λ1 , λ2 ) does not exist. We have the following. 2

Corollary 7 A GDD.(3, n, n + 1, 4; 2s, 2t + 1) does not exist.

5 GDD(3, n, n + 1, 4; λ1 , λ2 ) with Small Parameters 5.1 GDD(3, 3, 4, 4; λ1 , λ2 ) If .n = n1 = 3, then .λ1 < r1 , i.e., a GDD.(3, 3, 4, 4; λ1 , λ2 ) does not exist if .λ1 = r1 . 2 Using the bounds for .λ1 obtained in Sect. 4.1, we have . 11λ 20 ≤ λ1 < 7λ2 . Also, 2λ1 +7λ2 .r1 = r2 = implies that all elements of the same group occur together, and 3 as block size is greater than 3, we have second associate pairs from .G1 and .G2 . The upper bound on .λ1 can be improved further. Suppose .G1 = {a, b, c}, .G2 = {x, y, z}, and a GDD.(3, 3, 4, 4; λ1 , λ2 ) exists. Since .r1 > λ1 , let .r1 = λ1 + s where 2λ1 +7λ2 .s = r1 − λ1 = − λ1 = 7λ23−λ1 . Notice that there are at most .λ1 − s blocks 3 of type .{a, b, c, ∗} and .{x, y, z, ∗} in the design, respectively, where .∗ is an element from a different group. This implies that the second associate pairs of elements from .G1 and .G2 appear at most .2 × 3 × (λ1 − s) times in these blocks. Since .G3 has four elements, there are at most .4λ2 blocks of the type .{a, b, c, ∗} where .∗ is an element in .G3 , and there are at most .4λ2 blocks of the type .{x, y, z, ∗} where .∗ is an element in .G3 . In the remaining of the blocks of the type .{a, b, c, ∗} or .{x, y, z, ∗} in the design, .∗ must be an element from .G2 or .G1 , respectively. That is, the second associate pairs of elements from .G1 and .G2 appear at most .6(λ1 − s) − 2 × 3 × 4λ2 times in those remaining blocks. Since the second associate pairs of elements from .G1 and .G2 appear .9λ2 times in the design, .6(λ1 − s) − 24λ2 ≤ 9λ2 , i.e., .2(λ1 − s) ≤ 2 11λ2 . Replacing s with . 7λ23−λ1 and simplify, we have .λ1 ≤ 47λ 8 . Note that from the necessary conditions, a GDD.(3, 3, 4, 4; λ1 , 2t + 1) does not exist. GDD.(3, 3, 4, 4; λ1 , 2) If .n = 3 and .λ2 = 2, from the necessary conditions, 47 we have .λ1 ≡ 2 (mod 3) and . 11 10 ≤ λ1 ≤ 4 , i.e. .λ1 = 2, 5, 8 or 11. A GDD.(3, 3, 4, 4; 2, 2) is just a BIBD.(10, 4, 2) which exists. The blocks of a GDD.(3, .3, 4, 4; λ1 , 2) where .λ1 = 5, 8, 11, respectively, are given below. Let .G1 = {1, 2, 3}, .G2 = {a, b, c} and .G3 = {w, x, y, z}.

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Example 6 A GDD.(3, 3, 4, 4; 5, 2): 1 2 . 3 a

1 2 3 b

1a 2b 3c c1

a b . x y

a b z w

a b c 3

1 2 x y

1 2 z w

1 3 x z

1 3 y w

2 3 x w

a c y w

b c x w

b c y z

x y z w

x y z w

x y z w

a b c 2

a c x z

2 3 y z

Example 7 A GDD.(3, 3, 4, 4; 8, 2): 1 2 . 3 x a b . x y

1 2 3 y

a b z w

1 1 2 2 3 3 zw

1 2 3 x

1 2 3 y

1 2 3 z

1 2 3 w

a b c 1

a c x z

b c x w

b c y z

x y z w

x y z w

x y z w

1 2 3 z

1 2 3 w

a b c x

a b c y

a b c z

x y z w

x y z w

x y z w

x y z w

x y z w

a c y w

a b c 2

a b c 3

a b c 1

a b c 2

a b c 3

x y z w

x y z w

x y z w

x y z w

a b c w

a b c x

a b c y

a b c z

a b c w

x y z w

x y z w

x y z w

Example 8 A GDD.(3, 3, 4, 4; 11, 2): 1 2 . 3 x 1 2 . 3 a

1 2 3 b

1 2 3 y

1 2 3 c

1 1 2 2 3 3 zw

a b c 1

a b c 2

a b c 3

1 2 3 x

1 2 3 y

x y z w

x y z w

x y z w

Lemma 12 Necessary conditions are sufficient for the existence of a GDD.(3, 3, 4, 4; λ1 , 2). GDD.(3, 3, 4, 4; λ1 , 4) If .n = 3 and .λ2 = 4, from the necessary conditions, we 47 have .λ1 ≡ 1 (mod 3) and . 11 5 ≤ λ1 ≤ 2 , i.e. .λ1 = 3t + 1 where .1 ≤ t ≤ 7.

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A GDD.(3, 3, 4, 4; 4, 4) is just a BIBD.(10, 4, 4) which exists. A GDD .(3, 3, 4, 4; 7, 4) can be obtained by combining the blocks of a BIBD.(10, 4, 2) and the blocks of a GDD.(3, 3, 4, 4; 5, 2). A GDD.(3, 3, 4, 4; 10, 4) can be obtained by putting together two copies of the blocks of a GDD.(3, 3, 4, 4; 5, 2). A GDD.(3, 3, 4, 4; 13, 4) can be obtained by combining the blocks of a GDD .(3, 3, 4, 4; 5, 2) and the blocks of a GDD.(3, 3, 4, 4; 8, 2). A GDD .(3, 3, 4, 4; .16, 4) can be obtained by putting together two copies of the blocks of a GDD.(3, 3, 4, 4; 8, 2). A GDD.(3, 3, 4, 4; 19, 4) can be obtained by combining the blocks of a GDD .(3, 3, 4, 4; 8, 2) and the blocks of a GDD .(3, 3, 4, 4; 11, 2). A GDD .(3, 3, 4, 4; .22, 4) can be obtained by putting together two copies of the blocks of a GDD .(3, 3, 4, 4; 11, 2). .

Lemma 13 Necessary conditions are sufficient for the existence of a GDD.(3, 3, 4, 4; λ1 , 4). GDD.(3, 3, 4, 4; λ1 , 6) If .n = 3 and .λ2 = 6, from the necessary conditions, we 141 have .λ1 ≡ 0 (mod 3) and . 33 10 ≤ λ1 ≤ 4 , i.e. .λ1 = 3t where .2 ≤ t ≤ 11. A GDD.(3, 3, 4, 4; 6, 6) can be obtained by combining three copies of the blocks of a GDD.(3, 3, 4, 4; 2, 2) (it is also a BIBD.(10, 4, 6) which exists). A GDD.(3, 3, 4, 4; 9, 6) can be obtained by combining two copies of the blocks of a GDD.(3, 3, 4, 4; 2, 2) and the blocks of a GDD.(3, 3, 4, 4; 5, 2). A GDD.(3, 3, 4, 4; 12, .6) can be obtained by combining two copies of the blocks of a GDD.(3, 3, 4, 4; 5, 2) and the blocks of a GDD.(3, 3, 4, 4; 2, 2). A GDD.(3, 3, 4, 4; 15, 6) can be obtained by combining three copies of the blocks of a GDD.(3, 3, 4, 4; 5, 2). A GDD.(3, 3, 4, 4; .18, 6) can be obtained by combining two copies of the blocks of a GDD.(3, 3, 4, 4; 5, .2) and the blocks of a GDD.(3, 3, 4, 4; 8, 2). A GDD.(3, 3, 4, 4; 21, 6) can be obtained by combining two copies of the blocks of a GDD.(3, 3, 4, 4; 5, 2) and the blocks of a GDD.(3, 3, 4, 4; 11, 2). A GDD.(3, 3, 4, 4; 24, 6) can be obtained by combining three copies of the blocks of a GDD.(3, 3, 4, 4; 8, 2). A GDD.(3, 3, 4, 4; 27, 6) can be obtained by combining two copies of the blocks of a GDD.(3, 3, 4, 4; 11, .2) and the blocks of a GDD.(3, 3, 4, 4; 5, 2). A GDD.(3, 3, 4, 4; 30, 6) can be obtained by combining two copies of the blocks of a GDD.(3, 3, 4, 4; 11, 2) and the blocks of a GDD.(3, 3, 4, 4; 8, 2). A GDD.(3, 3, 4, 4; 33, 6) can be obtained by combining three copies of the blocks of a GDD.(3, 3, 4, 4; 11, 2). We have the following. Lemma 14 Necessary conditions are sufficient for the existence of a GDD.(3, 3, 4, 4; λ1 , 6).

5.2 GDD(3, 4, 5, 4; λ1 , λ2 ) If .n = 4, from the necessary conditions, we have .λ1 ≡ 0 (mod 3), .λ2 ≡ 0 (mod 3), 2 and .λ1 and .λ2 have the same parity. In addition, . 47λ 95 ≤ λ1 ≤ 9λ2 .

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GDD.(3, 4, 5, 4; 3(3 + t), 3(1 + t)) If .λ1 = 3(3 + t) and .λ2 = 3(1 + t) for t ≥ 0, the necessary conditions are satisfied. We first construct the 52 blocks of a GDD.(3, 4, 5, 4; 9, 3) (where .t = 0) as follows. Suppose .G1 = {x, y, z}, .G2 = {a, b, c, d} and .G3 = {1, 2, 3, 4, 5}: The first 28 blocks consist of .{a, b, c, i}, .{a, b, d, i}, .{a, c, d, i}, .{b, c, d, i} where .i ∈ G1 , 3 copies of .{a, b, c, d}, .{x, y, z, i} where .i ∈ G3 , .{1, 2, x, y}, .{3, 4, x, y}, .{1, 3, x, z}, .{2, 4, x, z}, .{1, 4, y, z}, .{2, 3, y, z}, and 2 copies of .{x, y, z, 5}. A BIBD.(5, 3, 6) on .G3 is 3- resolvable in four 3- resolvable classes. The next 20 blocks are formed by combining the triples of the first class with a, of the second with b, of the third with c and the fourth with d. The last four blocks are the blocks of a BIBD.(5, 4, 3) on .G3 except for .{1, 2, 3, 4}. A BIBD.(12, 4, 3) exists (it is also a GDD.(3, 4, 5, 4; 3, 3)). Combining the blocks of t copies of a BIBD.(12, 4, 3) and the blocks of a GDD.(3, 4, 5, 4; 9, 3), we have a GDD.(3, 4, 5, 4; 9 + 3t, 3 + 3t) for all .t ≥ 0. .

Example 9 A GDD.(3, 4, 5, 4; 9, 3): a b . c x

a b d x

a c d x

b c d x

a b c y

a b d y

a c d y

b c d y

a b c z

a b d z

a c d z

b c d z

a b c d

a b c d

a b c d

x y z 1

x y z 2

x y z 3

x y . z 4

x y z 5

1 2 x y

3 4 x y

1 3 x z

2 4 x z

1 4 y z

2 3 y z

x y z 5

x y z 5

1 2 3 a

1 4 5 a

1 3 4 a

2 3 5 a

2 4 5 a

1 2 4 b

1 2 5 b

1 3 5 b

2 3 . 4 b

3 4 5 b

1 2 3 c

1 4 5 c

1 3 4 c

2 3 5 c

2 4 5 c

1 2 4 d

1 2 5 d

1 3 5 d

2 3 4 d

3 4 5 d

11 22 34 55

1 3 4 5

2 3 4 5

5.3 GDD(3, 6, 7, 4; λ1 , λ2 ) 2 If .n = 6, from the necessary conditions, we have .λ2 − λ1 ≡ 0 (mod 6) and . 27λ 65 ≤ λ1 ≤ 13λ2 .

GDD.(3, 6, 7, 4; λ1 , 1) and GDD.(3, 6t, 6t + 1, 4; 12t + 1, 1) If .n = 6 and .λ2 = 1, from the necessary conditions, .λ1 = 7 and 13. Let .G1 = {a, b, c}, .G2 = {x, y, z, w, u, v} and .G3 = {1, 2, 3, 4, 5, 6, 7}. The 59 blocks of a GDD.(3, 6, 7, 4; 7, 1) are as follows: .G1 ∪ {i} where

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i ∈ G2 , .{ab12, ac34, .bc56, a756, b734, c712}, the blocks from a BIBD.(13, 4, 1) on .G2 ∪ G3 , the blocks of a BIBD.(6, 4, 6) on .G2 , and the blocks in .{7135, 7135, 7136, 7145, 7146, 7234, 7245, 7246, 7236, 7256, 1245, 1246, 1235, .1236, 1436, 1456, 2345, 2356, 3456}. Let .G1 = {a, b, c}, .G2 = {x1 , x2 , · · · , x6t } and .G3 = {y1 , y2 , · · · , y6t+1 }. The blocks of a GDD.(3, 6t, 6t + 1, 4; 12t + 1, 1) are as follows: .G1 ∪ {i} where .i ∈ G2 , .G1 ∪ {j } where .j ∈ G3 , the blocks of a BIBD.(v, 4, 1) where .v ∈ G2 ∪ G3 , the blocks of a BIBD.(v, 4, 12t) where .v ∈ G2 , and the blocks of a BIBD.(v, 4, 12t) where .v ∈ G3 . Specifically, if .t = 1, we have a GDD.(3, 6, 7, 4; 13, 1). We have the following lemma. .

Lemma 15 The necessary conditions of a GDD.(3, 6, 7, 4; λ1 , 1) are sufficient.

6 Summary In this chapter we proposed and studied two new problems of group divisible designs with three groups and block sizes.The first problem involved group sizes 1, n and .n + 2, respectively. We showed that if an equal number of blocks of block type configurations was required, a GDD.(1, n, n + t, 4; λ1 , .λ2 ) where .t ≥ 0 did not exist (in the Appendix). We then focused our study on the case where .t = 2 where there was no block configuration restriction. We provided the necessary conditions for the existence of a GDD.(1, n, n+2, 4; λ1 , .λ2 ). Also, we proved that the necessary conditions were sufficient for the existence of a GDD.(1, n, n + 2, 4; λ1 , λ2 ) when .λ1 ≥ λ2 and .λ1 ≡ λ2 (mod 6) except for n even, .λ1 ≡ 0 (mod 6) and .λ2 ≡ 3 (mod 6), and for .n ≡ 2 (mod 6), λ1 ≡ 2, 4 (mod 6) and .λ2 ≡ 1, 5 (mod 6). The second problem involved group sizes 3, n and .n + 1, respectively. Group size of the first group being 3 made the problem much more challenging. We obtained necessary conditions for the existence of a GDD.(3, n, n + 1, 4; λ1 , .λ2 ), and we provided constructions for certain families of the problem and showed that the necessary conditions were sufficient for these families.

Appendix: Non-Existence Result of GDD(1, n, n + t, 4; λ1 , λ2 ) with Configuration Restrictions As the first group is of size 1, these GDDs, if they exist, must have blocks of type (1,3) or .(1, 1, 2). Suppose a GDD.(1, n, n (+)t, 4; λ1(, λ2)) exists where .t ≥ 1. Then the number of t (t−1) 2 first associate pairs equals . n2 λ1 + n+t 2 λ1 = .(n + (t − 1)n + 2 )λ1 and the number of second associate pairs equals .(n + (n + t) + n(n + t))λ2 . As the block size is 4, there are four types of blocks for a GDD.(1, n, n + t, 4, λ1 , λ2 ): blocks of

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67

configuration .(0, 4), .(1, 1, 2), .(1, 3) and .(2, 2) respectively. Let .b1 , .b2 , .b3 and .b4 be the number of blocks of type .(0, 4), .(1, 1, 2), .(1, 3) and .(2, 2) respectively. Then there are .6b1 + b2 + 3b3 + 2b4 first associate pairs in the design. Hence, 6b1 + b2 + 3b3 + 2b4 = (n2 + (t − 1)n +

.

t (t − 1) )λ1 . 2

In addition, there are .5b2 + 3b3 + 4b4 second associate pairs in the design. Hence, 5b2 + 3b3 + 4b4 = λ2 (n2 + (t + 2)n + t).

.

(1)

Since the point of .G1 can be paired with any one element of .G2 and .G3 , there are .λ2 (2n + t) second associate pairs containing the point of .G1 in the design. Let .r1 be the number of blocks containing the point of .G1 . Then there are .3r1 second associate pairs containing the point of .G1 in the design. Hence, .r1 = λ2 (2n+t) . 3

Equal Number of Blocks of Type (1, 1, 2) and (2, 2) Now, we establish the non-existence results for GDD.(1, n, n + t, 4; λ1 , λ2 ) .(n ≥ 2) with equal number of blocks having configuration .(1, 1, 2) and .(2, 2) only. Suppose a GDD.(1, n, n+2, 4; λ1 , λ2 ) with equal number of blocks of configuration .(1, 1, 2) and .(2, 2) exists and has b blocks, so .b2 = b4 = b2 . Then from Eq. (1), we have .b = 2λ2 (n +(t+2)n+t) . 9 Since the point of .G1 must occur in . b2 blocks of configuration .(1, 1, 2) but does not occur in any blocks of configuration .(2, 2), .r1 = b2 . Then . λ2 (2n+t) = 3 2

λ2 (n2 +(t+2)n+t) . 9

This implies .n2 + (t − 4)n − 2t = 0 which has integer solutions only when .n = 3 and .t = 3. For this case, .b = 6λ2 , and counting the number of first associate pairs, .b = 12λ1 . Hence .λ2 = 2λ1 . Now the number of first associate pairs from the third group is .15λ1 . As each block has at most one first associate pair from a group, .15λ1 must be less than or equal to b. Hence for nonzero .λ1 , a GDD.(1, 3, 6, 4; λ1 , λ2 ) does not exist. Hence, we have the following theorem. Theorem 8 A GDD.(1, n, n + t, 4; λ1 , λ2 ) with equal number of blocks having configuration .(1, 1, 2) and .(2, 2) respectively does not exist.

Equal Number of Blocks of Type (1, 1, 2) and (0, 4) Here we establish non-existence results for GDD.(1, n, n+t, 4; λ1 , λ2 ) .(n ≥ 2) with equal number of blocks having configuration .(1, 1, 2) and .(0, 4) only.

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Suppose a GDD.(1, n, n + t, 4; λ1 , λ2 ) with equal number of blocks of configuration .(1, 1, 2) and .(0, 4) exists and has b blocks. Then from Eq. (1), we have 2λ2 (n2 +(t+2)n+t) .b = . 5 Since the point of .G1 must occur in . b2 blocks of configuration .(1, 1, 2) but does not occur in any block of configuration .(0, 4), we must have .r1 = b2 . Then .

= r1 = b2 = λ2 (n +(t+2)n+t) . This implies .3n2 + (3t − 4)n − 2t = 0, but 5 this does not have an integer solution. Hence, we have the following theorem. 2

λ2 (2n+t) 3

Theorem 9 A GDD.(1, n, n + t, 4; λ1 , λ2 ) with equal number of blocks having configuration .(1, 1, 2) and .(0, 4) respectively does not exist.

Equal Number of Blocks of Type (0, 4), (1, 1, 2) and (2, 2) First, we establish non-existence results for GDD.(1, n, n + 2, 4; λ1 , λ2 ) .(n ≥ 2) with equal number of blocks having configuration .(0, 4), .(1, 1, 2) and .(2, 2) only. Suppose a GDD.(1, n, n + 2, 4; λ1 , λ2 ) with equal number of blocks having configuration .(0, 4), .(2, 2) and .(1, 1, 2) exists and has b blocks. Then from Eq. (1), 2 we have .b = 3λ2 (n +4n+2) . This implies b is a multiple of 3 and satisfies the 9 requirement for having equal number of blocks of configuration .(0, 4), .(1, 1, 2) and .(2, 2) respectively. Since the point of .G1 occurs only in blocks of configuration .(1, 1, 2), .r1 = b3 . = λ2 (n +4n+2) . This implies .n2 − 2n − 4 = 0, which has no integer Then . λ2 (2n+2) 3 9 solution. Thus, we have the following theorem, 2

Theorem 10 A GDD.(1, n, n + 2, 4; λ1 , λ2 ) with equal number of blocks of configuration .(0, 4), .(2, 2) and .(1, 1, 2) respectively does not exist. Now we reproduce similar result for GDD.(1, n, n + t, 4; λ1 , λ2 ) .(n ≥ 2) with equal number of blocks having configuration .(0, 4), .(1, 1, 2) and .(2, 2) only. Suppose a GDD.(1, n, n + t, 4; λ1 , λ2 ) with equal number of blocks having configuration .(0, 4), .(2, 2) and .(1, 1, 2) exists and has b blocks.Then from Eq. (1), 2 we have .b = 3λ2 (n +(t+2)n+t) . 9 Since the point of .G1 occurs only in blocks of configuration .(1, 1, 2), .r1 = b3 .

= λ2 (n +(t+2)n+t) . This implies .n2 + (t − 4)n − 2t = 0 which has an Then . λ2 (2n+t) 3 9 integer solution only when .n = 3 and .t = 3. If the design GDD.(1, 3, 6, 4; λ1 , λ2 ) exists, it has .b = 9λ2 blocks and .r1 = 3λ2 . Now counting the number of first associate pairs in these blocks , if the design exists, we get .18λ1 = .27λ2 . Hence .b = 6λ1 . As per the condition, .2λ1 first associate pairs of the group of size three (second group) are in the blocks of type .(2, 2) and hence the remaining .λ1 = 3t first associate pairs of the second group must come in the blocks of type .(1, 1, 2). This means that the 6t second associate pairs of first and second group elements 2

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have occurred in these .λ1 blocks already and hence it is impossible to construct the remaining .(1, 1, 2) blocks. Thus, we have the following theorem, Theorem 11 A GDD.(1, n, n + t, 4; λ1 , λ2 ) with equal number of blocks of configuration .(0, 4), .(2, 2) and .(1, 1, 2) respectively does not exist.

References 1. Colbourn, C. J., Rosa, A.: Triple systems. Oxford Science Publications, Clarendon Press, Oxford (1999). 2. Fu, H. L., Rodger, C. A.: Group divisible designs with two associate classes: n = 2 or m = 2. J. Combin. Theory Ser. A, 83, 94–117 (1998). 3. Fu, H. L., Rodger, C. A., Sarvate, D. G.: The existence of group divisible designs with first and second associates having block size three. Ars Combin., 54, 33–50 (2000). 4. Hanani, H.: The existence and construction of balanced incomplete block designs. Ann. Math. Stat., 32, 361–386 (1961). 5. Henson, D., Hurd, S. P., Sarvate, D. G.: Group divisible designs with three groups and block size four. Discrete Math., 307, 1693–1706 (2007). 6. Hurd, S. P., Mishra, N., Sarvate, D. G.: Group divisible designs with two groups and block size five with fixed block configuration. J. Combin. Math. Combin. Comput., 70, 15–31 (2009). 7. Hurd, S. P., Sarvate, D. G.: Odd and even group divisible designs with two groups and block size four. Discrete Math., 284, 189–196 (2004). 8. Hurd, S. P., Sarvate, D. G.: Group divisible designs with two groups and block configuration (1, 4). J. Comb. Inf. and Syst Sci., 32, 1–4 (2007). 9. Hurd, S. P., Sarvate, D. G.: Group divisible designs with block size four and two groups. Discrete Math., 308, 2663–2673 (2008). 10. Keranen, M. S., Laffin, M. R.: Fixed block configuration group divisible designs with block size six. Discrete Math., 308, 2663–2673 (2008). 11. Mullin, R. C., Gronau, H. D. O. F.: PBDs and GDDs: The Basics, the CRC Handbook of Combinatorial Designs (edited by Colbourn, C. J. and Dinitz, D. H., pp. 185–213). CRC Press, Boca Raton, FL (1996). 12. Mwesigwa, R., Sarvate, D. G., Zhang, L.: Group divisible designs of four groups and block size five with configuration (1, 1, 1, 2). J. Algebra Comb. Discrete Struct. Appl., 3(3), 187– 194 (2016). 13. Namyalo, K., Sarvate, D. G., Zhang, L.: GDD(n1 , n, n+1, 4; λ1 , λ2 ) : n1 = 1 or 2. J. Combin. Math. Combin. Comput., 110, 19–37 (2019). 14. Ndungo, I., Sarvate, D. G.: GDD(n, 2, 4; λ1 , λ2 ) with equal number of even and odd blocks. Discrete Math., 339, 1344–1354 (2016). 15. Punnim, N., Sarvate, D. G.: A construction for group divisible designs with two groups. Congr. Numer., 185, 57–60 (2007). 16. Mishra, N., Namyalo, K., Sarvate, D. G.: Group divisible designs with block size five from clatworthy’s table. Commun. Stat. - Theory and Methods, 47(9), 2085–2097 (2018). 17. Nanfuka, M., Sarvate, D. G.: Group divisible designs with block size 4 and number of groups 2 or 3. Ars Combin., 141, 229–241 (2018). 18. Sarvate, D. G., Zhang, L.: Group divisible designs of three groups and block size five with configuration (1, 2, 2). Australas. J. Combin., 66(2), 333–343 (2016). 19. Street, A. P., Street, D. J.: Combinatorics of Experimental Design. Clarendon Press, Oxford (1987).

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20. Street, A. P., Street, D. J.: Partially Balanced Incomplete Block Designs, the CRC Handbook of Combinatorial Designs (edited by Colbourn, C. J. and Dinitz, D. H., pp. 419–423). CRC Press, Boca Raton, FL (1996). 21. Zhu, M., Ge, G.: Mixed group divisible designs with three groups and block size four. Discrete Math., 310, 2323–2326 (2010).

A New Absolute Irreducibility Criterion for Multivariate Polynomials over Finite Fields Carlos Agrinsoni, Heeralal Janwa, and Moises Delgado

Abstract One of the key problems in algebraic geometry and its applications in coding theory, cryptography, and other disciplines is to determine whether a variety defined by a set of polynomials is absolutely irreducible; i.e., it remains irreducible in the algebraic closure of the defining field. The famous Eisenstein criterion for irreducibility works only over the defining fields. One important place where this is needed is when one applies the Riemann-Roch theorem. Another important application is in bounding the number of rational points or exponential sums through the Weil conjectures. In this chapter, we consider the hypersurfaces defined by generalized trinomials. We present a new absolute irreducibility criterion for generalized trinomials over finite fields. Our criterion does not require testing for irreducibility in the ground field or in any extension field. We just require multivariate GCD computations and the square-free property. Since almost all polynomials are known to be square-free, our absolute irreducibility criterion proves the absolute irreducibility of almost all generalized trinomials. Keywords Hypersurfaces · Generalized trinomials · Finite fields

C. Agrinsoni (□) Department of Mathematics, Purdue University, West Lafayette, IN, USA e-mail: [email protected] H. Janwa College of Natural Sciences Department of Mathematics, University of Puerto Rico, Rio Piedras, San Juan, Puerto Rico M. Delgado University of Puerto Rico, Cayey, Cayey, Puerto Rico © The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 S. Heuss et al. (eds.), Combinatorics, Graph Theory and Computing, Springer Proceedings in Mathematics & Statistics 462, https://doi.org/10.1007/978-3-031-62166-6_5

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1 Introduction A fundamental problem in algebra and algebraic geometry is testing the irreducibility (respectively, absolute irreducibility) of a polynomial. Finding practical criteria for irreducibility and absolute irreducibility is vital for applications in pure and applied mathematics. In the case of polynomials in one variable, the Eisenstein criterion is a classical method to test irreducibility [13]. This criterion has been generalized to test the irreducibility (and in some cases, absolute irreducibility) of multivariate polynomials using Newton polygons, prime ideals, and valuations (see, for example, [10, 30, 35]). The absolutely irreducible property is required in many applications in different areas of mathematics such as algebraic geometry, combinatorics [41], coding theory [23, 24, 40], cryptography [18], finite geometry [19, 20], function field sieve [1], permutation polynomials [34], and scattered polynomials [5]. Several fundamental results in algebraic geometry, such as the bounds of Weil, Deligne, and Bombieri on the number of rational points and on exponential sums require the absolute irreducibility of the underlying variety ( see [11], for singular curves, see [4]). The following concrete applications are of particular relevance to our results. In coding theory and cryptography, substantial advances have been made to prove the exceptional almost perfect nonlinear (APN) conjecture by proving that a certain variety, defined by the corresponding multivariate polynomial, is absolutely irreducible or contains an absolutely irreducible component [18]. When defining an algebraic geometric code, an important requirement is that the underlying curve be absolutely irreducible [21]. Algebraic geometric codes typically use bivariate polynomials in applications. Beelen and Pellikaan [7] characterize many families of absolutely irreducible bivariate polynomials. In finite geometry, the Segre-Bartocci conjecture was settled by proving that a multivariate polynomial contains an absolutely irreducible polynomial [19]. In this chapter, we will consider hypersurfaces defined by generalized trinomials. There do exist a few criteria for testing the absolute irreducibility of a polynomial. The method of Newton polygons generalizes the Eisenstein criterion in two different ways: (1) Eisenstein-Dumas criterion [12, 44]; (2) the Stepanov criterion (sometimes called the Stepanov-Schmidt criterion) [37–39]. An even more general criterion uses Newton polytopes [14]. However, these algorithms are often not effective in proving theorems. The test of absolute irreducibility that we present here is a good test for finding good absolutely irreducible bivariate polynomials. A highly effective method to test absolute irreducibility was given by Janwa et al. [23, 24] using Bezout’s theorem and intersection multiplicities. This method had been used to settle the Segre-Bartocci conjecture [19], and the exceptional APN conjecture for monomials [18, 23–25], and reducing the general exceptional APN conjecture to exponents involving only the Gold, Kasami-Welch and even degree cases. We also prove results on the existence of an absolutely irreducible factor over the defining field. Such a factor guarantees a sufficient number of rational points by

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the bounds of Weil, Deligne, Bombieri, and Lachaud-Ghorpade (see [3, 24]). The criterion of [3, 24] has applications to the exceptional APN conjecture, to the SegreBartocci conjecture [19], and to the exceptional planar conjecture [6]. Aubry et al. [3] reduced such problems from three dimensions to two dimensions by hyperplane intersections with applications to the general exceptional APN function conjecture. One can devise algorithms to test the absolute irreducibility of a polynomial. For details, see Gao [15], Kaltofen [28], von zur Gathen [42] and Heintz and Sieveking [17]. Another possibility is to design an algorithm to factor polynomials. Berlekamp developed the first efficient algorithm to factor polynomials in one variable over finite fields [8, 9]. Later, Lenstra et al. [33] designed a polynomial-time algorithm to factor univariate polynomials over rational numbers, while Kaltofen generalized it for multivariate polynomials [26, 27, 29]. Lenstra gave the first algorithm to factor multivariate polynomials in polynomial time [32] over finite fields via Hensel lifting. Weinberger, assuming the Riemann hypothesis, proved that there exist polynomialtime algorithms to determine the number of factors of univariate polynomials with rational coefficients [45]. In finite fields, the factors of an irreducible polynomial are conjugates, and the number of factors divides the degree of the polynomial [31], as stated in Lemma 2. Recently, in Agrinsoni et al. [2], we gave a bound on the number of factors a polynomial with a square-free leading form could have. Our proof is based on the concept of the degree gap of a polynomial (see the definition below [2].) This chapter is organized as follows. In Sect. 2, we provide some background materials. In Sect. 3, we present a theorem that gives sufficient conditions for a generalized trinomial to be absolutely irreducible. In the last section, we convert our criteria into an algorithm and explore its time complexity.

2 Background Recall that a multivariate polynomial, .G(X) in the polynomial ring .Fq (X) := Fq [X1 , . . . , Xn ], is called absolutely irreducible if it remains irreducible over any extension of .Fq . Furthermore, in the graded degree homogeneous decomposition (in the decreasing degree order), .G(X) decomposes as G(X) = Gd (X) + Gd1 · · · + Gdm (X),

.

(1)

where .Gdi (X) is a non-zero homogeneous polynomial of degree .di , .1 ≤ i ≤ m, and .d > d1 > · · · > dm . In the rest of the paper, from here on, polynomials will be written in decreasing degree homogeneous forms. .Gdm := TG (X) := T (X) is called the tangent cone of G. In the case when .n = 2, the linear factors of .TG (X) in the algebraic closure of .Fq are called the tangent lines of G. In addition, the degree of .G(X) is defined as the degree of .Gd (X). In the two-variable case, we have the following recent result.

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Lemma 1 (Bartoli and Schmidt[6]) Let .H (x, y) ∈ Fq [x, y] and suppose that the tangent cone of H contains a reduced linear factor over .Fq . Then H has an absolutely irreducible factor defined over .Fq . For possible degrees of absolutely irreducible factors, we also have. Lemma 2 (Kopparty and Yekhanin [31]) If .p(X) ∈ Fq [X1 , . . . , Xn ] is of degree d and is irreducible in .Fq [X1 , . . . , Xn ]. Then there exists r with .r | d and an absolutely irreducible polynomial .h(X) ∈ Fq r [X1 , . . . , Xn ] of degree .d/r such that p(X) = c

⨅

.

σ (h(X)),

σ ∈G

where .G = Gal(Fq r /Fq ) and .c ∈ Fq . Furthermore, if .p(X) is homogeneous, then so is .h(X). The following result about trinomials is known. In this chapter, we will show the absolute irreducibility of many more trinomials. Proposition 1 (Beelen and Pellikaan [7]) Let .F (x, y) = αXa + βXb Y c + γ Y d , where .a, b, c, and d are nonnegative integers, and .α, β, and .γ are nonzero elements of an algebraically closed field .F. Assume that .(a, 0), .(b, c) and .(0, d) are three distinct points. Then .F (X, Y ) is reducible if and only if .ac + bd = ad or the characteristic of .F divides the exponents .a, b, c, and d. We recently introduced the degree-gap of a polynomial [2]. We showed that every factor of a multivariate polynomial .F (X) with a square-free leading form has degree-gap greater than or equal to that of the polynomial .F (X). Definition 1 Let .F (X) ∈ Fq [X1 , . . . , Xn ] be a polynomial of degree m with at least two terms. We define the degree-gap .DG(F ) as the difference .d − d1 from Eq. (1). If .F (X) is a homogeneous polynomial, then .DG(F ) is defined as infinity. Theorem 1 (Agrinsoni et al. [2]) Let .F (X) = Fm (X)+H (X). If .Fm (X) is squarefree, then every factor of .F (X) has degree-gap at least that of .F (X). For an integer .n ≥ 1, let .T (n, q) denote the set of all polynomials in .Fq [x, y] of total degree n that are monic in x and have degree n in x. Gao and Lauder [16] gave bounds for the number of square-free polynomials and reducible polynomials in .T (n, q). Let .t (n, q) = |T (n, q)|. Proposition 2 (Gao and Lauder [16]) Let .r(n, q) be the number of reducible polynomials in .T (n, q). Then for .n ≥ 6, .

3 r(n, q) 1 4 1 ≤ · ≤ · n−1 . 4 q n−1 t (n, q) 3 q

The following proposition bounds the number of polynomials that are not squarefree.

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Proposition 3 (Gao and Lauder [16]) Let .r0 (n, q) be the number of polynomials in .T (n, q) that are not square-free. Then for .n ≥ 5, .

3 r0 (n, q) 1 4 1 ≤ · ≤ · 2n−1 . 4 q 2n−1 t (n, q) 3 q

Proposition 3 implies that almost all polynomials are square-free and we can conclude at least the following result. Proposition 4 (Gao and Lauder [16]) Most multivariate polynomials are squarefree.

3 Results Definition 2 Let F (X) = F (X1 , . . . , Xn ) ∈ Fq [X1 , . . . , Xn ]. We define F (X) to be a generalized trinomial if F (X) = Fa (X) + Fb (X) + Fc (X), where a /= b /= c, Fa (X) (respectively Fb (X), Fc (X)) is a homogeneous polynomial of degree a (respectively degree b and degree c), and Fa (X), Fb (X), Fc (X) ∈ Fq [X1 , . . . , Xn ]. Unlike Lemma 1, we prove the following result on absolute irreducibility that does not require showing the irreducibility of multivariate polynomials over any extension fields. We just need GCD and square-freeness. Testing square-freeness and GCD is simpler than showing irreducibility over field extensions. Theorem 2 Let F (X) = Fd (X)+Fd−e (X)+Fd−a (X) ∈ Fq [X]. If Fd (X) is squarefree, (Fd , Fd−e , Fd−a ) = 1 and e ⍿ a, then F (X) is absolutely irreducible. Proof Suppose that F = (Ps + Ps−1 + · · · + P0 )(Qt + Qt−1 + · · · + Q0 ),

.

(2)

where Pi (resp. Qj ) is either a form of degree i (resp. j ) or zero. Now we have Fd = Ps Qt .

.

(3)

Equating the forms of degree d − 1, we obtain Fd−1 = Ps Qt−1 + Ps−1 Qt .

.

(4)

Note that e ⍿ a and e /= 1. Then Fd−1 = 0, and Eq. (4) can be rewritten as follows: Ps Qt−1 = −Ps−1 Qt .

.

(5)

Hence, Ps | Ps−1 Qt and Qt | Ps Qt−1 . By Eq. (3) and Fd (X) being square free, we have (Ps , Qt ) = 1. Therefore, the divisibilities imply that Ps | Ps−1 , and Qt | Qt−1 . Thus, Ps−1 = Qt−1 = 0.

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Similarly, equating the forms of degree d − 2, d − 3, . . . , d − (e − 1), we get, recursively, Ps−j = Qt−j = 0,

.

(6)

for j ∈ {1, 2, . . . , e − 1}. By Eq. (2), we have Fd−e = Ps Qt−e + Ps−e Qt .

.

Let m = ⎣ ae ⎦. Assume that m ≥ 2. For 2 ≤ k ≤ m, we prove by induction that the following statement is true: S(k) : “For j ∈ {1, 2, . . . , ke − 1} and e ⍿ j , Ps−j = Qt−j = 0.”

.

(7)

The Basic Step We show that S(k) is true for k = 2. By Eq. (6), we have Ps−j = Qt−j = 0 for j ∈ {1, 2, . . . , e − 1}. Let j ∈ {e + 1, e + 2, . . . , 2e − 1}. Then, for the given generalized trinomial, Fd−j = 0 =

j Σ

.

Ps−i Qt−j +i =

i=0

Ps Qt−j +

e−1 Σ

.

Ps−i Qt−j +i +

j −1 Σ

Ps−i Qt−j +i + Ps−j Qt .

i=e

i=1

Using Eq. (6), we conclude that 0 = Ps Qt−j + Ps−j Qt

.

and Ps Qt−j = −Ps−j Qt .

.

Therefore, Ps−j = Qt−j = 0 for j ∈ {e + 1, . . . , ke − 1} (recall that (Ps , Qt ) = 1). Induction Step Assume that S(j ) is true for all j , 2 ≤ j ≤ k − 1 < m. We will show that S(k) is true. Let j ∈ {(k − 1)e + 1, (k − 1)e + 2, . . . , ke − 1}. Then Fd−j = 0 =

j Σ

.

i=0

Ps−i Qt−j +i =

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Ps Qt−j +

(k−1)e−1 Σ

.

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Ps−i Qt−j +i +

Ps−ie Qt−j +ie

i=1

i=1,e⍿i j −1 Σ

+

k−1 Σ

Ps−i Qt−j +i + Ps−j Qt .

(8)

i=(k−1)e+1

By the induction hypothesis, Σ P Q = 0, as Ps−i = 0 for i ∈ {1, 2, . . . (k − 1)e − 1}, e ⍿ i. 1. (k−1)e i=1,e⍿i s−i t−j +i Σk−1 2. i=1 Ps−ie Qt−j +ie = 0, as Qt−j +ie = 0 for i ∈ {e, 2e, . . . , (k − 1)e}, (as j − ie ∈ {1, 2, . . . (k − 1)e − 1} \ {e, 2e, . . . , (k − 1)e}). Σj −1 By Eq. (6), i=(k−1)e+1 Ps−i Qt−j +i = 0 as Qt−j +i = 0 for i ∈ {(k − 1)e + 1, . . . , j − 1}, (as j − i ∈ {1, . . . , e − 1}). Now Eq. (8) can be rewritten as Fd−j = 0 =

j Σ

.

Ps−i Qt−j +i = Ps Qt−j + Ps−j Qt .

i=0

This implies that Ps Qt−j = −Ps−j Qt .

.

Hence, Ps−j = Qt−j = 0 for j ∈ {(k − 1)e + 1, (k − 1)e + 2, . . . , ke − 1}, as (Ps , Qt ) = 1. Therefore, S(k) is true. Let j ∈ {me + 1, me + 2, . . . , a − 1}, then Fd−j = 0 =

j Σ

.

Ps−i Qt−j +i =

i=0

Ps Qt−j +

me−1 Σ

.

Ps−i Qt−j +i +

i=1,e⍿i

+

j −1 Σ

m Σ

Ps−ie Qt−j +ie

i=1

Ps−i Qt−j +i + Ps−j Qt .

i=me+1

By our inductive proof, Σ 1. me−1 i=1,e⍿i Ps−i Qt−j +i = 0, as Ps−i = 0 for i ∈ {1, 2, . . . me − 1}, e ⍿ i.

(9)

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Σm

i=1 Ps−ie Qt−j +ie = 0, as Qt−j +ie = 0 for i ∈ {e, 2e, . . . , me}, ( as j − ie ∈ {1, 2, . . . me − 1} \ {e, 2e, . . . , me}). Σj −1 By Eq. (6), i=me+1 Ps−i Qt−j +i = 0 as Qt−j +i = 0 for i ∈ {me + 1, . . . , j − 1}, (as j − i ∈ {1, . . . , e − 1}). Now Eq. (9) can be rewritten as

2.

Fd−j = 0 =

j Σ

.

Ps−i Qt−j +i = Ps Qt−j + Ps−j Qt .

i=0

This implies that Ps Qt−j = −Ps−j Qt .

.

Hence Ps−j = Qt−j = 0 for j ∈ {me + 1, me + 2, . . . , a − 1}. Therefore, we obtain Fd−a =

d−a Σ

.

Ps−i Qt−a+i

i=0

.

= Ps Qt−a +

a−1 Σ

Ps−j Qt−a+j +

a−1 Σ

Ps−j Qt−a+j + Ps−a Qt .

(10)

j =1, e|j

j =1, e⍿j

Σa−1

Ps−j Qt−a+j = 0 since Ps−j = 0 for every j ∈ {i | 1 ≤ i ≤ Σ a − 1, e ⍿ i} and a−1 j =1, e|j Ps−j Qt−a+j = 0 since Qt−a+j = 0 for every j ∈ {i | 1 ≤ i ≤ a − 1, e ⍿ i}. (We note that the assumption e ⍿ a is crucial here to prove that many of the terms are zero.) Therefore, Eq. (10) can be simplified as Now

j =1, e⍿j

Fd−a = Ps Qt−a + Ps−a Qt .

.

(11)

The tangent cone of F (X) is the product of the tangent cone of P and Q, because the lowest degree form of a polynomial is the product of the lowest degree forms of its factors. Therefore, there exist b and c such that Fd−a = Ps−b Qt−c ,

.

(12)

and Ps−i = 0 for i > b and Qt−j = 0 for j > c. Since Fd−a is the tangent cone of F (X), we have that Fd−a is the product of the tangent cones of P and Q. Therefore, either Ps−a = 0 or Qt−a = 0. If Ps−a = 0, then by Eq. (12), the tangent cone of P (X) is Ps . This contradicts (Fd , Fd−e , Fd−a ) = 1. Similarly, if Qt−a = 0, then by Eq. (12) the tangent cone of Q(X) is Qt . This contradicts (Fd , Fd−e , Fd−a ) = 1. ⨆ ⨅

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Corollary 1 If F (X) = Fd (X) + Fd−e (X) + Fd−a (X) ∈ Fq [X], where Fd (X) is square free, and e ⍿ a, then F (X) can be written as F (X) = G(X)H (X), where G(X) ∈ Fq [X] is a homogeneous polynomial and H (X) ∈ Fq [X] is an absolutely irreducible generalized trinomial. Furthermore, G(X) = (Fd (X), Fd−e (X), Fd−a (X)). Remark To demonstrate the importance of Corollary 1, we observe that in many applications, such as the existence or nonexistence of an exceptional APN, an exceptional permutational polynomial and other exceptional objects, one only needs the existence of an absolutely irreducible factor of a multivariate polynomial. These results are then derived using the existence of enough rational points over a finite field by applying Weil, Deligne, Lang-Weil, Ghorpade-Lachaud, and other bounds. We can conjecture along the lines of Proposition 4 that most generalized trinomials have an absolutely irreducible factor.

4 Algorithm Algorithm 1: Absolute irreducibility testing Data: F (X) = Fd (X) + Fd−e (X) + Fd−a (X), a generalized trinomial in Fq [X]; Result: The polynomial is absolutely irreducible or the analysis is not conclusive if Fd (X) is square free then if (Fd , Fd−e , Fd−a ) = 1 then if e ⍿ a then return(F (X) is absolutely irreducible) else return(Inconclusive) end else return(F (X) is reducible.) end else return(Inconclusive) end

Time Complexity of Algorithm Let f = O(deg(F ), q) be the time complexity of finding the GCD of multivariate polynomials and let g = O(deg(F ), q) be the time complexity of checking the square-freeness. Then the time complexity of our algorithm is just O(f + g). In the case when F (X) is a bivariate polynomial, we dehomogenize Fd (X) into a one variable polynomial. The time complexity of this algorithm can be reduced to only the time complexity of the square-freeness test plus the time complexity of GCD of polynomials of one variable. Several results are known about fast GCD computations of single- or multivariate polynomials over finite fields. They vary from using the Euclidean algorithm to computations with Groebner bases (we refer to [22, 36, 43]). Most absolute

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irreducibility testing algorithms require irreducibility testing over the ground field and over many extension fields (which can be indeterminate). Compared to those algorithms, our algorithm just requires square-free testing and GCD computations a much simpler task. Acknowledgments We thank the referee for a very thorough reading of the manuscript and for helpful suggestions. Carlos Agrinsoni was supported by NASA Training Grant No. NNX15AI11H and 80NSSC20M0052. Opinions, findings, conclusions, or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of NASA.

References 1. L. M. Adleman. The function field sieve. In Algorithmic number theory (Ithaca, NY, 1994), volume 877 of Lecture Notes in Comput. Sci., pages 108–121. Springer, Berlin, 1994. 2. C. Agrinsoni, H. Janwa, and M. Delgado. New absolute irreducibility testing criteria and factorization of multivariate polynomials. In F. Hoffman, S. Holliday, Z. Rosen, F. Shahrokhi, and J. Wierman (eds.) Combinatorics, graph theory and computing, pages 403–412. Springer International Publishing, Cham, 2024. 3. Y. Aubry, G. McGuire, and F. Rodier. A few more functions that are not APN infinitely often. In Finite fields: theory and applications, volume 518 of Contemp. Math., pages 23–31. Amer. Math. Soc., Providence, RI, 2010. 4. Y. Aubry and M. Perret. A Weil theorem for singular curves. In Arithmetic, geometry and coding theory (Luminy, 1993), pages 1–7. de Gruyter, Berlin, 1996. 5. D. Bartoli and M. Montanucci. Towards the full classification of exceptional scattered polynomials. arXiv preprint arXiv:1905.11390, 2019. 6. D. Bartoli and K.-U. Schmidt. Low-degree planar polynomials over finite fields of characteristic two. Journal of Algebra, 535:541–555, 2019. 7. P. Beelen and R. Pellikaan. The newton polygon of plane curves with many rational points. Designs, Codes and Cryptography, 21(1/3):41–67, 2000. 8. E. R. Berlekamp. Factoring polynomials over finite fields. Bell System Tech. J., 46:1853–1859, 1967. 9. E. R. Berlekamp. Factoring polynomials over large finite fields. Math. Comp., 24:713–735, 1970. 10. J. W. S. Cassels. Local fields, volume 3 of London Mathematical Society Student Texts. Cambridge University Press, Cambridge, 1986. 11. P. Deligne. La conjecture de Weil. i. Publications Mathématiques de l’Institut des Hautes Études Scientifiques, 43(1):273–307, 1974. 12. G. Dumas. Sur quelques cas d’irréductibilité des polynômes à coefficients rationnels. Journal de Mathématiques Pures et Appliquées, 2:191–258, 1906. 13. G. Eisenstein. Über die Irreductibilität und einige andere Eigenschaften der Gleichung, von welcher die Theilung der ganzen Lemniscate abhängt. J. Reine Angew. Math., 39:160–179, 1850. 14. S. Gao. Absolute irreducibility of polynomials via Newton polytopes. J. Algebra, 237(2):501– 520, 2001. 15. S. Gao. Factoring multivariate polynomials via partial differential equations. Math. Comp., 72(242):801–822, 2003. 16. S. Gao and A. Lauder. Hensel lifting and bivariate polynomial factorisation over finite fields. Mathematics of Computation, 71(240):1663–1676, 2002.

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17. J. Heintz and M. Sieveking. Absolute primality of polynomials is decidable in random polynomial time in the number of variables. In Automata, languages and programming (Akko, 1981), volume 115 of Lecture Notes in Comput. Sci., pages 16–28. Springer, Berlin-New York, 1981. 18. F. Hernando and G. McGuire. Proof of a conjecture on the sequence of exceptional numbers, classifying cyclic codes and APN functions. J. Algebra, 343:78–92, 2011. 19. F. Hernando and G. McGuire. Proof of a conjecture of Segre and Bartocci on monomial hyperovals in projective planes. Des. Codes Cryptogr., 65(3):275–289, 2012. 20. J. W. P. Hirschfeld. Projective geometries over finite fields. The Clarendon Press, Oxford University Press, New York, 1979. Oxford Mathematical Monographs. 21. T. Høholdt, J. H. Van Lint, and R. Pellikaan. Algebraic geometry codes. Handbook of coding theory, 1(Part 1):871–961, 1998. 22. J. Hu and M. Monagan. A fast parallel sparse polynomial gcd algorithm. In Proceedings of the ACM on International Symposium on Symbolic and Algebraic Computation, pages 271–278, 2016. 23. H. Janwa, G. M. McGuire, and R. M. Wilson. Double-error-correcting cyclic codes and absolutely irreducible polynomials over GF(2). J. Algebra, 178(2):665–676, 1995. 24. H. Janwa and R. M. Wilson. Hyperplane sections of Fermat varieties in P3 in char. 2 and some applications to cyclic codes. In Applied algebra, algebraic algorithms and error-correcting codes (San Juan, PR, 1993), volume 673 of Lecture Notes in Comput. Sci., pages 180–194. Springer, Berlin, 1993. 25. D. Jedlicka. APN monomials over GF(2n ) for infinitely many n. Finite Fields Appl., 13(4):1006–1028, 2007. 26. E. Kaltofen. A polynomial reduction from multivariate to bivariate integral polynomial factorization. In Proceedings of the fourteenth annual ACM symposium on Theory of computing, pages 261–266, 1982. 27. E. Kaltofen. A polynomial-time reduction from bivariate to univariate integral polynomial factorization. In 23rd annual symposium on foundations of computer science (Chicago, Ill., 1982), pages 57–64. IEEE, New York, 1982. 28. E. Kaltofen. Fast parallel absolute irreducibility testing. J. Symbolic Comput., 1(1):57–67, 1985. 29. E. Kaltofen. Polynomial-time reductions from multivariate to bi- and univariate integral polynomial factorization. SIAM J. Comput., 14(2):469–489, 1985. 30. S. K. Khanduja and J. Saha. On a generalization of Eisenstein’s irreducibility criterion. Mathematika, 44(1):37–41, 1997. 31. S. Kopparty and S. Yekhanin. Detecting rational points on hypersurfaces over finite fields. In Twenty-Third Annual IEEE Conference on Computational Complexity, pages 311–320. IEEE Computer Soc., Los Alamitos, CA, 2008. 32. A. K. Lenstra. Factoring multivariate polynomials over finite fields. J. Comput. System Sci., 30(2):235–248, 1985. 33. A. K. Lenstra, H. W. Lenstra, and L. Lovász. Factoring polynomials with rational coefficients. Math. Ann., 261(4):515–534, 1982. 34. R. Lidl and H. Niederreiter. Finite fields, volume 20 of Encyclopedia of Mathematics and its Applications. Addison-Wesley Publishing Company, Advanced Book Program, Reading, MA, 1983. With a foreword by P. M. Cohn. 35. S. MacLane. The Schönemann-Eisenstein irreducibility criteria in terms of prime ideals. Trans. Amer. Math. Soc., 43(2):226–239, 1938. 36. T. Sasaki and M. Suzuki. Three new algorithms for multivariate polynomial gcd. Journal of symbolic computation, 13(4):395–411, 1992. 37. W. Schmidt. Equations over finite fields: an elementary approach. Kendrick Press, Heber City, UT, second edition, 2004. 38. S. A. Stepanov. Congruences with two unknowns. Izv. Akad. Nauk SSSR Ser. Mat., 36:683–711, 1972.

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On Combinatorial Interpretations of Some Elements of the Riordan Group Melkamu Zeleke and Mahendra Jani

Abstract Riordan arrays, equipped with Shapiro’s multiplication rule, form a group and this group provides an interesting algebraic framework to solve combinatorial problems. In this chapter, we provide combinatorial interpretations for Riordan arrays of the form .R = (g(z), zg(z)), where .g(z) is a generating function satisfying a functional equation .g(z) = 1 + z ∗ g(z)k , where k is a positive integer greater than or equal to 2. The combinatorial interpretations are then used to obtain the inverses of these elements of the Bell subgroup of the Riordan group explicitly in terms of powers of .g(z). Keywords K-Trees · Dyck paths · Riordan arrays · Riordan group

1 Introduction 1.1 The Riordan Group Riordan arrays gained attention after Shapiro, Getu, Woan, and Woodson published their seminal work on the Riordan group in 1991 [9]. The central concept in their work is the development of a group structure which unifies many themes in enumeration. One can use this concept to obtain significant results in enumerative combinatorics [2, 10, 11]. Let .Fn be the collection of formal power series of the form {f (x) ∈ R[[x]]|f (x) = an x n + an+1 x n+1 + · · · , where an /= 0}.

.

M. Zeleke (✉) · M. Jani Department of Mathematics, William Paterson University, Wayne, NJ, USA e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 S. Heuss et al. (eds.), Combinatorics, Graph Theory and Computing, Springer Proceedings in Mathematics & Statistics 462, https://doi.org/10.1007/978-3-031-62166-6_6

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M. Zeleke and M. Jani (1)

(1)

The Riordan group .R = F0 ×F1 is the set of pairs .(g(x), f (x)) of formal power series .g(x) ∈ F0 , with .g(0) = 1, and .f (x) ∈ F1 , with .f ' (0) = 1, equipped with the binary operation referred to as Shapiro’s multiplication rule (g, f ) · (u, v) = (gu(f ), v(f )).

.

The identity element is .(1, x) and the inverse of .(g, f ) is given by (g, f )

.

−1

⎛ =

⎞ 1 ¯ ,f , g(f¯)

where .f¯ is the compositional inverse of .f (x). To each element .(g(x), f (x)) ∈ R, we associate a matrix .M(g, f ), with entries in the ring R, by defining its .(n, k)-th term to be tn,k = [x n ]g(x)f (x)k , where n, k ≥ 0.

.

We call the matrix .M(g, f ) the Riordan array associated to .(g(x), f (x)) ∈ R. The Fundamental Theorem of Riordan arrays [8, 9] states that if .A(z) and .B(z) are the generating functions of the column vectors .A = (a0 , a1 , a2 , · · · )T and .B = (b0 , b1 , b2 , · · · )T , then M(g, f ) · A = B if and only if B(z) = g(z)A(f (z)).

.

One of the most important results in the study of the Riordan group and the associated Riordan arrays is what is known as Roger’s sequence characterization of entries of the array [10]. More precisely, Roger’s theorem states that an infinite lower triangular array .M(g, f ) = (tn,k )n,k∈{0,1,2,··· } is a Riordan array if and only if sequences .A = {a0 /= 0, a1 , a2 , · · · } and .Z = {z0 /= 0, z1 , z2 , · · · } exist such that for each .n, k ∈ {0, 1, 2, · · · }, tn+1,k+1 =

∞ ⎲

.

aj tn,k+j

j =0

= a0 tn,k + a1 tn,k+1 + a2 tn,k+2 · · · and tn+1,0 =

∞ ⎲

.

zj tn,j

j =0

= z0 tn,0 + z1 tn,1 + z2 tn,2 · · · . Similar sequence characterizations of some of the well known subgroups of the Riordan group have been obtained recently [6]. For example, the Riordan array

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M(g, f ) = (tn,k )n,k∈N is in the Bell subgroup

.

(1)

B = {(g(x), f (x)) ∈ R|f (x) = xg(x)} = {(g(x), xg(x))|g(x) ∈ F0 }

.

if and only if there is a sequence .C = {c0 , c1 , c2 , · · · } such that tn+1,k = tn,k−1 +

⎲

.

cj tn−j,k for n, k = 0, 1, 2, · · · .

j ≥0

With the exception of some work done on alternative characterizations of Riordan arrays by Renzo Sprugnoli et al. [4, 7, 11], the underlying theory of the Riordan group is still not well developed and there is plenty of work that needs to be done to develop this algebraic framework so that Riordan arrays can be utilized more in discovering and proving significant results in enumerative combinatorics and beyond.

1.2 Generalized Ordered Trees K-trees are introduced as a generalization of regular ordered trees [5, 12]. If K is any nonempty subset of .{2, 3, 4, · · · }, then a K-tree is obtained by designating an edge of a distinguished k-cycle, with .k ∈ K, as a root and repeatedly gluing other k-cycles to existing ones along an edge. If an internal edge has more than one child, we glue the cycles to the internal edge with diminishing size as shown in the Fig. 1. Edges or 2-cycles are glued to the midpoint of a side of a k-cycle (.k > 2). Let .C(z; xk1 , xk2 , xk3 , . . .) be the generating function that enumerates K-trees by total number of cycles where we use the .xki s to keep track of the number of .ki cycles in the mixed tree. Depending on whether the distinguished cycle is a 2-cycle, a 3-cycle, a 4-cycle, etc. we obtain a functional relation that .C(z; xk1 , xk2 , xk3 , . . .) satisfies to be C(z; xk1 , xk2 , xk3 , . . .) = 1 + zxk1 C k1 + zxk2 C k2 + zxk3 C k3 + · · ·

.

(1)

Letting .u = C(z; xk1 , xk2 , xk3 , . . .) − 1, we can rewrite the functional relation as u = z(xk1 (u + 1)k1 + xk2 (u + 1)k2 + · · · )

.

Fig. 1 An example of a consisting of five 3-cycles and one 4-cycle

.{3, 4}-tree

(2)

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Applying the Lagrange Inversion Formula [10] to (2) we get: 1 n−1 n [u ](xk1 (u + 1)k1 + xk2 (u + 1)k2 + · · · ) n ⎞ ⎛ ⎲ 1 n = [un−1 ] xk1 n1 . . .(u + 1)n1 k1 +... n , . . . n 1 n1 +n2 +...=n ⎞⎛ ⎛ ⎞ ⎲ 1 n n1 k1 + · · · + 1 = xk1 n1 . . . n n n k · · · + 1 , . . . 1 1 1 n +n +...=n

[zn ]u =

.

1

2

. We deduce from these that: Theorem 1 The number of K-trees consisting of .ni .ki -cycles, .i = 1, 2, · · · , m is ⎞⎛ ⎛ ⎞ 1 n n1 k1 + · · · + nm km + 1 . , n n1 , . . . , nm n1 k1 + · · · + nm km + 1

(3)

where .n = n1 + n2 + . . . + nm . . For example, the number of .{3, 4}-trees with five 3-cycles and one 4-cycle is .

⎛ ⎞⎛ ⎞ 1 5×3+1×4+1 6 = 11, 628. 5×3+1×4+1 6 5, 1

. If all cycles of a K-tree have the same number of sides, say k, we refer to it as a homogeneous k-tree. The number of such homogeneous k-trees consisting of n k-cycles is Cn,k =

.

⎛ ⎞ ⎛ ⎞ 1 nk + 1 1 kn = . nk + 1 n (k − 1)n + 1 n

. For example, the twelve 3-trees consisting of .n = 3 cycles are shown below (Fig. 2): Fig. 2 The twelve 3-trees consisting of .n = 3 cycles

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2 Riordan Arrays Obtained from the Bell Subgroup 2.1 Generalized Dyck Paths and k-Trees A Dyck path of length 2n is a path from .(0, 0) to .(2n, 0) with unit up .U = (1, 1) and unit down .D = (1, −1) steps which never goes below the x-axis. Dyck paths and their equivalent reformulations are fundamental objects in combinatorics and have been studied since the second half of the nineteenth century in connection with various problems. For example, the solution to Bertrand’s ballot problem [1] is obtained very easily if we reformulate it as a properly constrained Dyck path enumeration problem. One possible way to generalize Dyck paths is to consider paths that use unit up steps .U = (1, 1) and down steps .D = (1, 1 − k) of length .k − 1. We will refer to such paths as k-Dyck paths, following Cameron and McLeod [3]. If .g(z) is the ordinary generating function of the number of k-Dyck paths from .(0, 0) to .(kn, 0) which never go below the x-axis, then it is not difficult to see from the well known decomposition of such paths that .g(z) = 1 + z(g(z))k . This functional equation is the same as the one for homogeneous k-trees [5] and we have a natural bijection between the two objects. Given a k-tree consisting of n k-cycles, traverse the edges in order starting from the root edge in a counterclockwise direction. Record each step away from the root edge as a unit up step .U = (1, 1) and each step towards the root edge in the traversal as a down step .D = (1, 1 − k) of length .k − 1. The paths obtained from k-trees in this way never go below the x-axis since the counterclockwise traversal of each k-cycle gives .k − 1 unit up steps and a single down step of length .k − 1. When we have more than one k-cycle attached to an internal edge, we start the traversal from the k-cycle at the bottom and move up. This process is reversible and gives a one-to-one correspondence between k-trees consisting of n k-cycles and k-Dyck paths from .(0, 0) to .(kn, 0) which never go below the x-axis. For example, the twelve 3-Dyck paths of length 9 shown in Fig. 3 correspond to the twelve 3-trees in Fig. 2.

2.2 Riordan Arrays, k-Trees, and k-Dyck Paths Cameron and Mcleod [3] used the Riordan array ⎡

1 ⎢1 ⎢ ⎢ ⎢3 .(T (z), zT (z)) = ⎢ ⎢ 12 ⎢ ⎢ 55 ⎣ .. .

0 1 2 7 30 .. .

0 0 1 3 12 .. .

00 00 00 10 41 .. .. . .

⎤ ··· ···⎥ ⎥ ⎥ ···⎥ ⎥ ···⎥, ⎥ ···⎥ ⎦ .. .

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Fig. 3 The twelve 3-Dyck paths of length 9

to count 3-Dyck paths that end at height .m ≥ 0. Since there is a one-to-one correspondence between k-Dyck paths and k-trees, we looked at an interpretation of the columns of the Riordan array ⎡

.. .. .. . . ⎢ . 2 2 3 .(Ck (z), zCk (z)) = ⎢ Ck (z) zCk (z) z Ck (z) ⎣ .. .. .. . . .

⎤ .. . ⎥ ···⎥ ⎦, .. .

where the leftmost column is the number of k-trees with generating function .Ck (z) that satisfies the functional equation .Ck (z) = 1 + zCk (z)k . We observed that the subsequent columns of the Riordan array .D = (Ck (z), zCk (z)) represent a family of k-trees with a fixed path consisting of m k-cycles in which only the root edges and leftmost edge along the fixed path have subtrees. Clearly, the generating function of such families of k-trees is .zm Ck (z)m+1 , for .m ∈ {1, 2, 3, · · · }. For example, if we have a fixed path consisting of .m = 2 4-cycles, we can attach other unrestricted subtrees to the root edge, the edge connecting the two 4-cycles in the fixed path, and to the leftmost edge of the path as shown in Fig. 4. Since there is a natural bijection between k-trees and k-Dyck paths, we looked for subfamilies of k-Dyck paths that correspond to the columns of the array. To obtain these families, consider a family of k-trees with a fixed path consisting of m k-cycles in which only the root edges and leftmost edge along the fixed path have subtrees. Draw a k-Dyck path starting at .(0, 0) and ending at .(i1 k, 0) corresponding to the k-subtree attached to the root edge of the fixed path consisting of .i1 k-cycles. Then

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Fig. 4 The generating function of this family of 4-trees is .z2 C43 (z)

Fig. 5 A typical k-Dyck path ending at .(X, m), where .X = nk − m(k − 1)

draw a unit up step .U = (1, 1) from .(i1 k, 0) to .(i1 k + 1, 1) and add a k-Dyck path starting .(i1 k + 1, 1) and ending at .(i2 k, 1) corresponding to the k-subtree attached to the root edge of the fixed path at level one consisting of .i2 k-cycles. Repeating this construction .m + 1 times, we obtain a k-Dyck path that starts at .(0, 0), end at height m, and never go below the x-axis as shown in the Fig. 5. This process is reversible and the following theorem follows. Theorem 2 There is a one to one correspondence between a family of k-trees with a fixed path consisting of m k-cycles in which only the root edges and leftmost edge along the fixed path have subtrees and a family of k-Dyck paths with unit up steps .U = (1, 1) and down steps .D = (1, 1 − k) that start from .(0, 0), ends at .(X, m), where .X = nk − m(k − 1), and never goes below the x-axis. Furthermore, the generating function of such families of k-trees or k-Dyck paths is .zm Ckm+1 (z). Extracting the coefficient of .zn , for each .n ≥ 0, from .zm Ckm+1 (z) using the Lagrange Inversion Formula as shown in (2), we obtain the following. Corollary 3 The number of k-Dyck paths that use unit up steps .U = (1, 1) and down steps .D = (1, 1 − k) starting from .(0, 0) and ending at .(nk − (k − 1)m, m) is dn,m =

.

⎛ ⎞ m+1 (n − m)k + m + 1 . (n − m)k + m + 1 n−m

Replacing each up step .U = (1, 1) in a typical k-Dyck path starting at .(0, 0) and ending at .(kn, 0) by a unit east step .E = (1, 0) and each down step .D = (1, 1 − k) by a unit north step .N = (0, 1), we obtain a lattice path starting from .(0, 0) to .((k − 1)n, n) not crossing the line .(k − 1)y = x. For example, the 4-Dyck path

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Fig. 6 An example of bijection between 4-trees and 4-Dyck paths

Fig. 7 A lattice path corresponding to the 4-Dyck path in Fig. 6

corresponding to a 4-tree consisting of .n = 6 4-cycles shown in Fig. 6 is equivalent to the lattice path starting from .(0, 0) and ending at .(18, 6) never crossing the line .3y = x shown in Fig. 7. Having a fixed path of length .m = 2 in the 4-tree is equivalent to the 4-Dyck path ending at height .m = 2 or .(18, 2), and this translates to the lattice path ending at .(14, 4). Hence, the above corollary can be restated for lattice paths as follows. Corollary 4 The number of lattice paths that use unit east steps .E = (1, 0) and unit north steps .N = (0, 1) not crossing the line .(k − 1)y = x starting from .(0, 0) and ending at .((k − 1)n − (k − 2)m, n − m) is dn,m =

.

⎛ ⎞ m+1 (n − m)k + m + 1 . (n − m)k + m + 1 n−m

Remark If we are given a lattice path from .(0, 0) to .(a, b) consisting of unit east E = (1, 0) and unit north .N = (0, 1) steps not crossing the line .(k − 1)y = x, then the number n of k-cycles in the k-tree with a fixed path of length m corresponding to this lattice path satisfies the equation:

.

.

⎞⎛ ⎞ ⎛ ⎞ ⎛ ⎞⎛ ⎞ ⎛ ⎞ ⎛ a 1 2−k n n k − 1 −(k − 2) a . = ⇔ = b 1 −k + 1 m m 1 −1 b

Thus, the number of lattice paths from .(0, 0) to .(a, b) not crossing the line .(k − 1)y = x is .

⎛ ⎞ a + (1 − k)b + 1 a + b + 1 . a+b+1 b

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3 Inverses of Elements of the Bell Subgroup Consider the Riordan array .(1 − z, z(1 − z)). If we wish to find its inverse in the Riordan group, we need .z(1⎛− z).⎞This can be obtained easily by √letting .f (z) = z(1 − z) and noting that .f f (z) = z which leads to .f (z) = 1− 21−4z . ⎞ ⎛ 1 Now using the fact that .(g, f )−1 = , f we see that .(1 − z, z(1 − z))−1 = g(f ) ⎛ √ ⎞ √ 1− 1−4z 1− 1−4z = zC(z)), , where .C(z) is the generating function for (C(z), 2z 2 the Catalan numbers, which also counts ordered trees. Multiplying the Riordan array .(T (z), zT (z)), where .T (z) is the generating function of ternary trees, by the sequence of the Catalan numbers .{cn } = {1, 1, 2, 5, 14, 42, · · · }, we obtain ⎡

1 ⎢ 1 ⎢ ⎢ ⎢ 3 ⎢ . ⎢ 12 ⎢ ⎢ 55 ⎢ ⎢ 273 ⎣ .. .

0 1 2 7 30 143 .. .

0 0 1 3 12 55 .. .

0 0 0 1 4 18 .. .

0 0 0 0 1 5 .. .

⎤ ⎤ ⎡ ⎤ ⎡ 1 1 0 ··· ⎥ ⎢ ⎥ ⎢ 0 ···⎥ ⎥ ⎢ 1⎥ ⎢ 2 ⎥ ⎥ ⎥ ⎢ ⎥ ⎢ 0 ···⎥ ⎢ 2 ⎥ ⎢ 7 ⎥ ⎥ ⎥ ⎢ ⎥ ⎢ 0 · · · ⎥ · ⎢ 5 ⎥ = ⎢ 30 ⎥ . ⎥ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ 0 ···⎥ ⎥ ⎢ 14 ⎥ ⎢ 143 ⎥ ⎢ 42 ⎥ ⎢ 728 ⎥ 1 ···⎥ ⎦ ⎦ ⎣ ⎦ ⎣ .. .. .. . . .

Since the entries of the right hand side of the above equation are the same as the entries of the second column of the Riordan array .(T (z), zT (z)), we applied the Fundamental Theorem of Riordan Arrays and noticed that .T (z)C(zT (z)) = √ T 2 (z) ⇔ C(zT (z)) = T (z). In fact, this identity is true since .C(z) = 1− 2z1−4z , and √ 1 − 4zT (z) 2zT (z) √ ⇔ 2zT 2 (z) − 1 = − 1 − 4zT (z)

C(zT (z)) = T (z) ⇔ T (z) =

.

1−

⇔ T (z) = 1 + zT 3 (z), which is the functional equation of the generating function of ternary trees or 3trees. This identity can also be obtained combinatorially by considering an ordered tree consisting of n edges, and replacing subtrees of this ordered tree at non-root vertices by ternary subtrees in such a way that the total number of edges is n in the newly constructed tree. For example, for .n = 3, we start with the five ordered trees with three edges and obtain seven additional mixed trees as shown in Fig. 8.

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Fig. 8 Composition of ordered and ternary trees consisting of .n = 3 edges

Fig. 9 An example of a hybrid 4-tree consisting of 10 cycles

Now, using the identity .C(zT (z)) = T (z) we see that: .

(T (z), zT (z)) ∗ (1 − zC(z), z(1 − zC(z))) = (T (z) (1 − zT (z)C(zT (z))) , zT (z) (1 − zT (z)C(zT (z)))) ⎛ ⎛ ⎞⎞ = T (z) − zT 3 (z), z T (z) − zT 3 (z) = (1, z),

and this implies that .(1 − zC(z), z(1 − zC(z))) is the inverse of the Bell subgroup element .(T (z), zT (z)). The combinatorial construction used above consisting of ordered trees and ternary trees can be extended to .(k − 1) and k-trees to obtain what we refer to as hybrid k-trees. We construct hybrid k-trees consisting of .(k − 1) and k cycles using the following rules (Fig. 9): 1. Use only .(k − 1)-cycles at the root or level 0. 2. At levels .l ≥ 1, (a) Traverse the edges of cycles at level .l − 1 counterclockwise and attach only .(k − 1)-cycles to all non-root edges except the rightmost edge. (b) On the rightmost edge, attach a .(k − 1)-cycle or a subtree of k-cycles. Lemma 5 There is a one-to-one correspondence between hybrid k-trees consisting n cycles, and pure k-trees consisting of the same number of k-cycles. Proof Consider a k-tree consisting of n k-cycles. Replace each k-cycles at the root or level 0 by a .(k − 1)-cycle. At levels .l ≥ 1, traverse the non-root edges counterclockwise and replace the attached k-cycles by .(k − 1)-cycles except for the rightmost edge. If a subtree consisting of k-cycles is attached to the rightmost

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edge at level l, then attach the same subtree to the rightmost edge of the .(k − 1)cycle corresponding to the k-cycle at level .l − 1. If you have to attach both .(k − 1) and k-cycles to the rightmost edge of a .(k − 1)-cycle at level .l − 1, draw the subtree consisting of k-cycles on the top to avoid double counting. Repeat these steps until the last level of the given tree. This process is clearly reversible, and we conclude that there is a one-to-one correspondence between k-trees and hybrid k-trees consisting of the same total number of cycles (Figs. 10 and 11). ⨆ ⨅ Theorem 6 The generating function of k-trees .Ck (z) satisfies the identity Ck−1 (zCk (z)) = Ck (z) and (Ck (z), zCk (z))−1 ⎛ ⎛ ⎞⎞ k−2 k−2 = 1 − zCk−1 (z) , z 1 − zCk−1 (z) .

.

Proof From the construction rules of hybrid k-trees, its generating function is Ck−1 (zCk (z)), where .zCk (z) accounts for a subtree of a k-tree that is attached to

.

Fig. 10 An example of a one-to-one correspondence between a 4-tree and hybrid 4-tree

Fig. 11 4-trees consisting 3 cycles and their corresponding hybrid 4-trees

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the rightmost edge of a .(k − 1)-cycle. Hence, .Ck (z) = Ck−1 (zCk (z)) by Lemma 5. Using this identity and the definition of the Riordan group operation, we see that .

⎛ ⎛ ⎞⎞ k−2 k−2 (Ck (z), zCk (z)) ∗ 1 − zCk−1 (z) , z 1 − zCk−1 (z) ⎛ ⎞ ⎛ k−2 = Ck (z) · 1 − zCk (z) · Ck−1 (zCk (z)) , zCk (z) ⎛ ⎞⎞ k−2 · 1 − zCk (z) · Ck−1 (zCk (z)) ⎛ ⎞⎞ ⎛ k−2 k−2 = Ck (z) − zCk2 (z) · Ck−1 (zCk (z)) , z Ck (z) − zCk2 (z) · Ck−1 (zCk (z)) ⎛ ⎞⎞ ⎛ = Ck (z) − zCkk (z), z Ck (z) − zCkk (z) = (1, z) since k -trees satisfy the equation Ck (z) = 1 + zCkk (z). ⨆ ⨅

Remark Using the Lagrange inversion formula and the result obtained in Theorem6, we can obtain each entry .dn,m of the inverse of .(Ck (z), zCk (z)) as shown below. ⎛ ⎞m+1 k−2 dn,m = [zn ]zm (1 − zCk−1 (z)

.

⎛ ⎞m+1 k−2 = [zn−m ] (1 − zCk−1 (z) ⎞ ⎛ m+1 ⎞j ⎲ ⎛m + 1⎞⎛ k−2 −zCk−1 = [zn−m ] ⎝ (z) ⎠ j j =0

=

min (n−m,m+1) ⎲

(−1)j

⎛ ⎞ m + 1 n−m−j (k−2)j [z ]Ck−1 (z) j

(−1)j

⎛ ⎞ ⎛ ⎞ m+1 (k − 2)j (k − 1)(n − m) − j . j (k − 1)(n − m) − j n−m−j

j =0

=

min (n−m,m+1) ⎲ j =0

For example, ⎡

1 ⎢ 1 ⎢ ⎢ ⎢ 4 ⎢ .(C4 (z), zC4 (z)) = ⎢ 22 ⎢ ⎢ 140 ⎢ ⎢ 969 ⎣ .. .

0 1 2 9 52 340 .. .

0 0 1 3 15 91 .. .

0 0 0 1 4 22 .. .

0 0 0 0 1 5 .. .

0 0 0 0 0 1 .. .

⎤ ··· ···⎥ ⎥ ⎥ ···⎥ ⎥ ···⎥, ⎥ ···⎥ ⎥ ···⎥ ⎦ .. .

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and ⎡

(C4 (z), zC4 (z))−1

.

1 ⎢ −1 ⎢ ⎢ ⎢ −2 ⎢ −7 =⎢ ⎢ ⎢ −30 ⎢ ⎢ −143 ⎣ .. .

0 1 −2 −3 −10 −42 .. .

0 0 1 −3 −3 −10 .. .

0 0 0 1 −4 −2 .. .

0 0 0 0 1 −5 .. .

⎤ 0 ··· 0 ···⎥ ⎥ ⎥ 0 ···⎥ ⎥ 0 ···⎥. ⎥ 0 ···⎥ ⎥ 1 ···⎥ ⎦ .. .. . .

Acknowledgments The authors thank Lou Shapiro of Howard University for his helpful suggestions on the construction of some of the combinatorial objects used in this work. The first author acknowledges assigned release time for research from William Paterson University of New Jersey.

References 1. D. Andre, ´ Solution directe du probleme ´ resolu ´ par M. Bertrand, C. R. Acad. Sci. Paris, 105 (1887), 436–437. 2. P. Barry, Riordan Arrays: A Primer, http://www.lulu.com, 2017. 3. N. T. Cameron and J. E. Mcleod, Returns and Hills on Dyck Paths, J. Integer Seq., 19 (2016), Article 16.6.1. 4. T.X. He and R. Sprugnoli, Sequence characterization of Riordan arrays, Discrete Math., 309 (2009), 3962–3974. 5. M. Jani, R. G. Rieper, and M. Zeleke, Enumeration of K-trees and their applications, Ann. Comb., 6 (2002), 375–382. 6. S.T. Jin, A characterization of the Riordan Bell subgroup by C-sequences, Korean J. Math., 17 (2009), 147–154. 7. A. Lunzo, D. Merlini, M.A. Maron, and R. Sprugnoli, Identities Induced by Riordan Arrays, Linear Algebra Appl., 436 (2012), 631–647. 8. L. W. Shapiro, A Survey of the Riordan Group, http://users.dimi.uniud.it/~giacomo.dellariccia/ Table%20of%20contents/Shapiro2005.pdf. 9. L. W. Shapiro, S. Getu, W. J. Woan, and L.C. Woodson, The Riordan Group, Discret. Appl. Math., 34 (1991), 229–239. 10. R. Sprugnoli, An Introduction to Mathematical Methods in Combinatorics, http://www.dsi. unifi.it/~resp/handbook.pdf. 11. R. Sprugnoli, D. Merlini, and M.C. Verri, Combinatorial Sums and Implicit Riordan Arrays, Discrete Math., 308 (2008), 5070–5077. 12. M. Zeleke and M. Jani, k-trees and Catalan Identities, Congressus Numerantium, 165 (2003), 39–49.

Production Matrices of Double Riordan Arrays Dennis Davenport, Fatima Fall, Julian Francis, and Trinity Lee

Abstract A double Riordan array is an infinite lower triangular matrix, denoted by .(g; f1 , f2 ), where g, .f1 , and .f2 are generating functions. The coefficients of the generating function g gives the first column of the matrix, and the remaining columns are found by multiplying the previous column by alternating .f1 and .f2 . In other words, (g; f1 , f2 ) = (g, gf1 , gf1 f2 , gf1 2 f2 , gf12 f22 , . . . ).

.

This is the columns construction of a double Riordan array. We can determine the elements of a double Riordan array using A- and Z-sequences which gives a row construction of a double Riordan array, see ([2] and [5]). In this chapter we define the production matrix of a double Riordan array, and show how it can be used to determine the A- and Z-sequences. Keywords Riordan array · Double Riordan array · A-sequence · Z-sequence

1 Introduction In 1991, Shapiro, Getu, Woan, and Woodson introduced a group of infinite lower triangular matrices called the Riordan group, see [8]. The elements of the group are defined by two power series g and f , where the coefficients of g give the leftmost column and the ith column is given by the coefficients of .g · f i , for .i = 1, 2, 3, . . .

D. Davenport (✉) · F. Fall · J. Francis Howard University, Washington, DC, USA e-mail: [email protected]; [email protected]; [email protected] T. Lee King University, Bristol, TN, USA e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 S. Heuss et al. (eds.), Combinatorics, Graph Theory and Computing, Springer Proceedings in Mathematics & Statistics 462, https://doi.org/10.1007/978-3-031-62166-6_7

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Explicitly, the followingΣconstruction is used to build a Riordan array. Let .g(z) = Σ∞ ∞ k fk zk , with .g0 /= 0 and .f1 /= 0. Let .dn,k be the k=0 gk z and .f (z) =( k=1 ) k coefficient of .zn in .g(z) f (z) . Then .D = (d ( n,k )n,k≥0 )is a (Riordan ) array and an element of the Riordan group. We write .D = g(z), f (z) = g, f . Theorem 1 (Shapiro Arrays) ( ) ( et al. ) [8]; The Fundamental Theorem Σ of Riordan k and let A be Let . g(z), f (z) = g, f be a Riordan array. Let .A(z) = ∞ a z k k=0 the column vector .A = (a0 , a1 , a2 , · · · )T . Then .(g, f )A = g(z)A(f (z)). Theorem 2 (Shapiro et al. [8]) Let .(g, f ) and .(G, F ) be two Riordan arrays. Then the operation *, given by .(g, f ) ∗ (G, F ) = (g(z)G(f (z)), F (f (z))) is matrix multiplication which is an associative binary operation, .(1, z) is the identity element and the inverse of .(g, f ) is.( (1 ) , f ), where .f is the compositional inverse of f . g f

The Riordan Group comprises various noteworthy subgroups. One such subgroup is the Checkerboard Subgroup, defined as the set of all elements .(g, f ) where g is even (with a nonzero constant term) and f is odd. This designation is inspired by the visual resemblance of .(g, f ) to a checkerboard pattern. Additionally, a Riordan array is considered aerated if it exhibits alternating zeros. Consequently, all elements within the Checkerboard Subgroup are representative of aerated arrays. Example 1 The following is the Pascal array, where .g(z) = So,

1 1−z

and .f (z) =

⎤ 1 ⎥ ⎢1 1 ⎥ ⎢ ⎥ ⎢ ⎥ ⎢1 2 1 . (g(z), f (z)) = ⎢ ⎥. ⎢1 3 3 1 ⎥ ⎥ ⎢ ⎣1 4 6 4 1⎦ ... ⎡

To find the aerated array, we let .G(z) = g(z2 ) and .F (z) = ⎡ 1 ⎢0 1 ⎢ ⎢1 0 ⎢ ⎢0 2 ⎢ ⎢ ⎢1 0 . (G(z), F (z)) = ⎢ ⎢0 3 ⎢ ⎢1 0 ⎢ ⎢0 4 ⎢ ⎣1 0

f (z2 ) z .

⎤ 1 0 3 0 6 0 10

1 0 4 0 10 0

1 0 5 0 15 ...

1 0 6 0

⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥. ⎥ ⎥ ⎥ 1 ⎥ 01 ⎥ ⎥ 7 0 1⎦

So that

z 1−z .

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Given any Riordan array, every element of the array that is not in the leftmost column can be written as a linear combination of elements in the row that is directly above starting from the preceding column. In addition, every element in the zeroth column other than the zeroth element can be expressed as a linear combination of the elements from the row that is directly above, see [6] and [7]. Hence, we can construct Riordan arrays by using the rows of the matrix. The following theorem tells us how to construct a Riordan array using its rows. Theorem 3 Let .D = (dn,k ) be an infinite lower triangular matrix. Then D is a ∞ Riordan array if and only if there exists two sequences .A = {ai }∞ i=0 and .Z = {zi }i=0 with .a0 /= 0 and .z0 /= 0 such that dn+1,k+1 =

∞ Σ

.

aj dn,k+j ; k, n = 0, 1, 2, . . . .

(1)

zj dn,j ; n = 0, 1, 2, . . .

(2)

j =0

dn+1,0 =

∞ Σ j =0

∞ The sequences .{ai }∞ i=0 and .{zi }i=0 , respectively, are called the A-sequence and Zsequence of the Riordan array D. The following theorem, due Merlini et al. [6], shows us how to find the A- and Z-sequences of a Riordan array if we know the generating functions that determine the array. ( ) Theorem 4 Let .D = g(t), f (t) be a Riordan array. Let A be the generating function of the A-sequence and Z the generating function of the Z-sequence. Then

⎞ ⎛ t 1 1 A(t) = ( · 1− ( ) and Z(t) = ) , f (t) f (t) g f (t)

.

where .f is the compositional inverse of f . See [1] and [6] for more information about A- and Z-sequences of Riordan arrays. Another method of finding the A- and Z-sequences of Riordan arrays is using the production (or Stieltjes) matrix. The concept of production matrices was initially introduced by Deutsch, Ferrari, and Rinaldi in [4]. For a basic explanation and exploration of production matrices see [1]. Definition 1 Let .(g, f ) be a Riordan array. The production matrix or Stieltjes matrix P is given by: P = (g, f )−1 · (g, f ),

.

where .(g, f ) is the truncated Riordan array with the first row omitted.

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Theorem 5 Let .R = array. Then the production matrix P ) ( (g, f ) be a Riordan for R is of the form . Z A tA t 2 A . . . , where Z is the generating function for the Z-sequence and A is the generating function for the A-sequence of R.

2 Double Riordan: A and Z Sequences In a Riordan array, we use one multiplier function. Suppose alternating rules are used to generate an infinite matrix similar to a Riordan array. For example, suppose we are looking at Dyck paths with bicolored edges only at even heights. For this case, we have two rules; one for rows at even height and the other for rows at odd height. In general, the set of double Riordan arrays is not closed under multiplication. However, if we require that g be an even function and .f1 and .f2 odd, then there is an analog of The Fundamental Theorem of Riordan Arrays, which gives us a binary operation, see [3]. Σ Σ∞ 2k 2k+1 , and DefinitionΣ 2 Let .g(t) = 1 + ∞ k=1 g2k t , .f1 (t) = k=0 f1,2k+1 t ∞ 2k+1 .f2 (t) = , where .f1,1 /= 0 and .f2,1 /= 0 . Then the double k=0 f2,2k+1 z Riordan matrix (or array) of .g, .f1 and .f2 , denoted by .(g; f1 , f2 ), has column vectors .

(g, gf1 , gf1 f2 , gf12 f2 , gf12 f22 , · · · ),

The set of all aerated double Riordan matrices is denoted as .DR. Theorem 6 (The Fundamental Theorem of Double Riordan Let .g(t) = Σ∞ Σ∞ Σ∞ Arrays)2k+1 2k 2k+1 , and .f (t) = . 2 k=0 g2k t , .f1 (t) = k=0 f1,2k+1 t k=0 f2,2k+1 t Σ∞ Σ∞ 2k and .B(t) = 2k Case 1: If .A(t) = k=0 a2k t k=0 b2k t , and .A = T T Then (a0 , 0, a2 , 0, · · · ) and .B = (b0 , 0, b2 , 0, · · · ) ⎛ are column vectors. ⎞ √ f1 (z)f2 (z) . .(g, f1 , f2 )A .= B if and only if .B(z) = g(z)A Σ∞ Σ∞ 2k+1 2k+1 with = Case 2: If .A(t) = k=0 a2k+1 t √ and .B(t) k=0 b2k+1 t √ .(g, f1 , f2 )A = B, then .B(t) = g(t) f1 /f2 A( f1 (t)f2 (t)). Using the Fundamental Theorem of Double Riordan Arrays, we can define a binary operation on .DR. Definition 3 Let .(g, f1 , f2 ) and .(G, F1 , F2 ) be elements √ of .DR. √ √ .(g; f1 , f2 ) ∗ (G; F1 , F2 ) = (gG( f1 f2 ); f1 /f2 F1 ( f1 f2 ), Then: √ √ f2 /f1 F2 ( f1 f2 )). Theorem 7 .(DR, ∗) is a group. Where the matrix .(1; t, t) is the identity and √ ¯ t h/f ¯ 1 (h), ¯ t h/f ¯ 2 (h)) ¯ is the inverse of .(g; f1 , f2 ), where .h = f1 f2 and ((1/g(h); ¯ is the compositional inverse of h. .h

.

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Note that the Checkerboard Subgroup of the Riordan group is also a subgroup of DR, where .f1 = f2 . Recall that a Riordan array has one Z-sequence and one A-sequence. It is interesting to note that there were two known ways to row construct elements in .DR. One using two Z-sequences and one A-sequence and the other using one Zsequence and two A-sequences. In this chapter we give another found by using production matrices. One found by He is given in the following theorem, see [5]. .

Theorem 8 (He [5]) Let .D = (g; f1 , f2 ) = (dn,k ) be an infinite lower triangular matrix. Then D is a Double Riordan array if and only if there exists three sequences ∞ , .A = {a }∞ and .Z = {z }∞ with .a .A1 = {a1,i } 2 2,i i=0 i i=0 1,0 /= 0, .a2,0 /= 0, and i=0 .z0 /= 0 such that for each n dn,2k−1 =

∞ Σ

.

a1,j dn−1,2k+2(j −1) ; k = 1, 2, . . . ,

j =0

dn,2k =

∞ Σ

a2,j dn−2,2k+2(j −1) ; k = 1, 2, . . . , and

j =0

dn,0 =

∞ Σ

zj dn−2,2j .

j =0

The following corollary allows us to find the the A- and Z-sequences using the generating functions that define the Double Riordan Array. Σ 2k CorollaryΣ 1 Let .D = (g; f1 , f2 ) be anΣ element of .DR. Let .A1 (t) = ∞ k=0 a1,k t , ∞ ∞ 2k 2k .A2 (t) = respectively be the generating k=0 a2,k t , and .Z(t) = k=0 zk t √ functions for the .A1 −sequence, .A2 −sequence, and Z-sequence. Let .h = f1 f2 . Then A1 (t) =

.

A2 (t) = Z(t) =

f1 (h) h t2 h

2

⎞ ⎛ g(0) . · 1 − ) ( 2 g h h 1

The next theorem was proved by Branch, Davenport, Frankson, Jones, and Thorpe, see [2]. Theorem 9 Let .D = (g; f1 , f2 ) = (dn,k ) be an infinite lower triangular matrix. Then D is a Double Riordan array if and only if there exists three sequences .A = ∞ ∞ {ai }∞ i=0 , .Z0 = {z0,i }i=0 and .Z1 = {z1,i }i=0 with .a0 /= 0, .z0,0 /= 0, and .z1,0 /= 0

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such that for each n ∞ Σ

dn,k =

.

aj dn−2,k+2(j −1) ; k = 2, . . . ,

j =0 ∞ Σ

dn,1 =

z1,j dn−2,2j +1 , and

j =0 ∞ Σ

dn,0 =

z0,j dn−2,2j .

j =0

Σ 2k CorollaryΣ 2 Let .D = (g; f1 , f2 ) be an element of DR. Let .A(t) = ∞ k=0 ak t , Σ ∞ ∞ 2k 2k .Z0 (t) = respectively be the generating k=0 z0,k t , and .Z1 (t) = k=0 z1,k t √ functions for the A-sequence, .Z0 −sequence, and .Z1 −sequence. Let .h = f1 f2 . Then A(t) =

t2

.

h

2

⎞ ⎛ g(0) . · 1 − ( ) 2 g h h ⎞ ⎛ f1,1 h 1 Z1 (t) = . 1− (h)2 k(h)

Z0 (t) =

1

3 The Production Matrix Definition 4 Let (g; f1 , f2 ) be a double Riordan array. The production matrix of the first kind P1 is given by: P1 = (g; f1 , f2 )−1 · (g; f1 , f2 ),

.

where (g; f1 , f2 ) is the truncated double Riordan array with the first row omitted. Definition 5 Let (g; f1 , f2 ) be a double Riordan array. The production matrix of the second kind P2 is given by: P2 = (g; f1 , f2 )−1 · (g; f1 , f2 ),

.

where (g; f1 , f2 ) is the truncated double Riordan array with the first two rows omitted.

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The next theorem is the new row construction which was found using the production matrix of the first kind. Theorem 10 Let D = (g; f1 , f2 ) = (dn,k ) be an infinite lower triangular matrix. Then D is a Double Riordan array if and only if there exists three sequences A1 = ∞ ∞ {a1,i }∞ i=0 , A2 = {a2,i }i=0 and Z = {zi }i=0 with a1,0 /= 0, a2,0 /= 0, and z0 /= 0 such that for each n dn,2k−1 =

∞ Σ

.

a1,j dn−1,2k+2(j −1) ; k = 0, 1, 2, . . . ,

j =0

dn,2k =

∞ Σ

a2,j dn−1,2k+2(j −1) ; k = 0, 1, 2, . . . , and

j =0

dn,0 =

∞ Σ

zj dn−1,2j .

j =0

Σ 2k CorollaryΣ 3 Let D = (g; f1 , f2 ) be anΣ element of DR. Let A1 (t) = ∞ k=0 a1,k t , ∞ ∞ 2k 2k respectively be the generating A2 (t) = k=0 a2,k t , and Z(t) = k=0 zk t √ functions for the A1 −sequence, A2 −sequence, and Z-sequence. Let h = f1 f2 . Then A1 (t) =

.

A2 (t) =

t f1 h f1 (h)

h ⎞ ⎛ 1 g(0) 1 . Z(t) = · − ( ) f1 h g h f1

Theorem 11 Let D = (g; f1 , f2 ) be a double Riordan array. Then the production ) ( matrix of the second kind P2 for D is of the form Z0 zZ1 A zA z2 A . . . , where Z0 is the first Z-sequence, Z1 is the second Z-sequence, and A is the A-sequence for D. Proof For the proof we will use the fundamental theorem of double Riordan arrays. Note that the truncated matrix of the double Riordan array with the top two rows removed is ⎛ ⎞ g − g(0) gf1 − f11 z gf1 (f1 f2 ) g(f1 f2 )2 , , , ,... .(g; f1 , f2 ) = z2 z2 z2 z2 We divide by z2 because the top two rows are deleted.

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Now h = we get

√ f1 f2 , so h2 = f1 f2 and f2 =

h2 f1 .

If A is even, then by the F T DRA

√ (g; f1 , f2 )A = g(z)A( f1 (z)f2 (z)).

.

Recall that (g; f1 , f2 )

.

Since

g(z)−g(0) z2

⎛ .

−1

⎛ =

1

zh

zh

, , g(h) f1 (h) f2 (h)

⎞ .

is even, we get

1

zh

,

,

zh

⎞

g(h) f1 (h) f2 (h)

·

g(z) − g(0) 1 g(h) − g(0) = · 2 z g(h) (h)2 =

1 (h)2

·

g(h) − g(0)

g(h) ⎞ ⎛ 1 g(0) = · 1− (h)2 g(h)

By Corollary 2, Z0 =

.

⎞ ⎛ g(0) . 1− (h)2 g(h) 1

Thus the 0th column of the production matrix of the second kind is Z0 . Now if A is odd, then by the F T DRA, / (g; f1 , f2 )A = g(z)

.

Now

gf1 (z)−f11 z z2

(g; f1 , f2 )

.

f1 √ A( f1 (z)f2 (z)). f2

is odd so

−1

gf1 (z) − f1,1 z 1 · = · z2 g(h) = =

1 g(h) 1 g(h)

/ zh /

· / ·

f1 (h)

·

f2 (h) gf1 (h) − f1,1 h · z(h) (h)2

f2 (h) gf1 (h) − f1,1 h · f1 (h) (h)2 f2 (h) g(h)f1 (h) − f1,1 h · (h)2 f1 (h)

Production Matrices of Double Riordan Arrays

= = = =

1 g(h) 1 g(h) 1

105

·

┌ | | ⎦ /

· ·

h2 (h) f1 (h)

f1 (h) z2 f1

(h)2

z

·

g(h) f1 (h) z (h)2

·

·

g(h)f1 (h) − f1,1 h (h)2 g(h)f1 (h) − f1,1 h

·

(h)2

g(h)f1 (h) − f1,1 h (h)2

g(h)f1 (h) − f1,1 h

g(h)f1 (h) ⎞ ⎛ f1,1 h z 1− = g(h)f1 (h) (h)2 ⎞ ⎛ z f1,1 h = 1− (h)2 (g · f1 )(h)

So,

z (h)2

⎛ 1−

f1,1 h (g·f1 )(h)

⎞ is the first column of the production matrix of the second

kind. And by Corollary 2, Z1 =

.

1 (h)2

⎛ 1−

f1,1 h

⎞

(g · f1 )(h)

.

Thus, the first column of the production matrix of the second kind is zZ1 If k is any positive integer, then the generating function g(f1 f2 )k is even and k the generating ⎛function gf k be any non negative integer; then ⎞ 1 (f ⎛1 f2 ) is odd. Let ⎞ (g1 , f1 , f2 )−1

Since

g(f1 ,f2 )k z2

⎛ .

1

g(f1 ,f2 )k z2

=

1 , z(h) , z(h) g(h) f1 (h) f2 (h)

g(f1 ,f2 )k z2

is even, we get: ,

z(h)

,

z(h)

g(h) f1 (h) f2 (h)

⎞ ·

g(f1 f2 )k 1 g(h)(f1 (h)f2 (h))k = · 2 z g(h) (h)2 = = =

(f1 f2 (h)) (h)2 (h2 (h))k (h)2 z2k (h)2

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Since gf1 (f1 , f2 )k is odd (g1 , f1 , f2 )

.

−1

⎛ ·

gf1 (f1 f2 )k z2

⎞

⎛

⎞

gf1 (f1 , f2 )k z2 g(h) f1 (h) f2 (h) / zh 1 f2 (h) = · · g(h) f1 (h) zh 1

=

· = = = =

,

z(h)

,

z(h)

g(h)f1 (h)(f1 (h) · f2 (h))k (h)2 z

·

f1 (h)

f1 (h)(f1 (h) · f2 (h))k (h)2

z((f1 · f2 )(h))k (h)2 z · z2k (h)2 z2k+1 (h)2

.

Again by Corollary 2, the A-sequence of D is A(z) =

.

This completes the proof.

z2 (h(z))2

. ⨆ ⨅

The proof of the following theorem is similar to the proof of Theorem 8. Hence, it is omitted. We do point out that the truncated matrix with the first row removed has the form ⎞ ⎛ g − 1 gf1 gf1 (f1 f2 ) g(f1 f2 )2 , , , ,... . .(g; f1 , f2 ) = z z z z Theorem 12 Let D = (g; f1 , f2 ) be a double Riordan array. Then the production ) ( matrix of the first kind P1 for D is of the form Z A1 zA2 z2 A1 z3 A2 z4 A1 . . . , where Z is the Z-sequence, A1 is the first A-sequence, and A2 is the second Asequence for D.

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107

4 Combinatorial Example For a combinatorial example, consider Schröder paths with no level steps at odd height. See the below grid. 1 1

9

1

8

1

6

1 1

29

5

21

3

1

46

10

2

132 86

36

5

15

355 137

51

543 188

731

Arranging these numbers as a lower triangular array we get the following DR matrix. ⎡

⎤

1 0 1 2 0 1 0 3 0 1

⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ .D = ⎢ ⎢ 5 ⎢ ⎢ 0 ⎢ ⎢ 15 ⎢ ⎣ 0

0 5 10 0 0 21 36 0

0 6 0 29 ...

1 0 8 0

⎥ ⎥ ⎥ ⎥ ⎥ ⎥ .. ⎥ −1 ⎥ . ⎥ =⇒ D ⎥ ⎥ 1 ⎥ 01 ⎥ ⎥ 9 0 1⎦

⎡

0 ⎢2 ⎢ ⎢0 ⎢ ⎢5 ⎢ ⎢ ⎢ 0 .D = ⎢ ⎢ ⎢15 ⎢ ⎢0 ⎢ ⎢51 ⎢ ⎣0

⎤

⎡

1 ⎢ 0 ⎢ ⎢ −2 ⎢ ⎢ ⎢ 0 ⎢ =⎢ ⎢ 5 ⎢ ⎢ 0 ⎢ ⎢−13 ⎢ ⎣ 0

1 0 1 −3 0 0 8 0 −21

−5 0 19 0

⎤

1 0 3 0

1 0 5

1 0

10 0 36 0 137

0 21 0 86 0

6 0 29 0 132

1 0 8 0 46 0 ...

⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ..⎥ ⎥ 1 .⎥ ⎥ ⎥ 0 1 ⎥ ⎥ 9 0 1 ⎥ 0 11 0 1 ⎥ ⎥ 57 0 12 0 ⎦

1 0 −6 0 25

1 0 −8 0 ...

⎥ ⎥ ⎥ ⎥ ⎥ ⎥ .. ⎥ ⎥ . ⎥ ⎥ ⎥ 1 ⎥ 0 1 ⎥ ⎥ −9 0 1 ⎦

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⎡

2 0 5 0

⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ 15 .D = ⎢ ⎢ ⎢ 0 ⎢ ⎢ 51 ⎢ ⎢ 0 ⎢ ⎣188

⎤

0 3 0 10

1 0 5 0

1 0 6

0 36 0 137 0

21 0 86 0 355

0 29 0 132 0

1 0 8 0 46 0 235 ...

1 0 9 0 57 0

⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ..⎥ ⎥ 1 .⎥ ⎥ ⎥ 0 1 ⎥ 11 0 1 ⎥ ⎥ 0 12 0 ⎥ ⎥ 80 0 14 ⎦

Now we compute .P1 and .P2 . P1 = D −1 ∗ D,

.

P2 = D −1 ∗ D.

.

As a result we get, ⎤

⎡

0 1 ⎢2 0 1 ⎢ ⎢0 1 0 ⎢ ⎢−1 0 2 ⎢ ⎢ ⎢0 0 0 .P1 = ⎢ ⎢ 1 0 −1 ⎢ ⎢0 0 0 ⎢ ⎢−1 0 1 ⎢ ⎣0 0 0

1 0 1 1 0 0 2 0 0 0 −1 0 0 ...

⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ 1 ⎥ ⎥ ⎥ 01 ⎥ 101 ⎥ ⎥ 0201⎥ ⎥ 0010⎦

Z − seq : (0, 2, 0, −1, 0, 1, 0, −1, . . .)

.

A1 − seq : (1, 0, 1)

A2 − seq : (1, 0, 2, 0, −1, 0, 1, . . .)

Production Matrices of Double Riordan Arrays

109

⎤

⎡

201 ⎢0 3 0 1 ⎢ ⎢1 0 3 0 ⎢ ⎢0 1 0 3 ⎢ ⎢ ⎢0 0 1 0 .P2 = ⎢ ⎢0 0 0 1 ⎢ ⎢0 0 0 0 ⎢ ⎢0 0 0 0 ⎢ ⎣0 0 0 0

Z1 − seq : (2, 0, 1)

.

1 0 3 0 1 0 0 ...

1 01 301 030 103 010

⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ 1⎥ ⎥ 0⎥ ⎥ 3⎦

Z2 − seq : (3, 0, 1)

A − seq : (1, 0, 3, 0, 1)

Acknowledgments This research was made possible by the generous support of the National Security Agency (NSA), the Mathematical Association of America (MAA), and the National Science Foundation (NSF) grant DUE-1356481.

References 1. Barry, P., Riordan Arrays: A Primer, Logic Press, Raleigh, 2016. 2. Branch, D., Davenport, D., Frankson, S., Jones, J., Thorpe, G., A and Z Sequences for Double Riordan Arrays. Springer Proceedings in Mathematics and Statistics, Vol. 388 (2022), 33–46. 3. Davenport, D. E., Shapiro, L. W., Woodson, L. C., The Double Riordan Array. The Electronic Journal of Combinatorics 18 (2011), 1–16. 4. Deutsch, E., Ferrari, L., Rinaldi, S., Production matrices and Riordan arrays, Ann. Comb., 13 (2009), 65–85. 5. He, T-X, Sequence Characterizations of Double Riordan Arrays and Their Compressions, Linear Algebra and Its Applications 549 (2018), 176–202. 6. Merlini D., Rodgers, D. G., Sprugnoli, R., Verri, M. C., On some alternative characterizations of Riordan arrays. Can. J. Math 49 (1997), 301–320. 7. Rodgers, D. G., Pascal triangles, Catalan Numbers and renewal arrays. Discrete Math 22 (1978), 301–310. 8. Shapiro, L. W., Getu, S., Woan, W. and, Woodson, L. C., The Riordan Group. Discrete Applied Mathematics 34 (1991), 229–239.

The Existence of a Knight’s Tour on the Surface of Rectangular Boxes Shengwei Lu and Carl Yerger

Abstract A knight’s tour is a sequence of knight’s moves such that each square on the board is visited exactly once. In this chapter, we show that a closed knight’s tour exists on the surface of a rectangular box of any size. Our general algorithm is to concatenate the top and bottom faces of a box with its side faces. When general criteria are not satisfied, especially when the dimensions of the rectangular box are small, we devise some special techniques to cover these cases. Keywords Knight’s tour · Hamiltonian · Box

1 Introduction In chess, a knight is a piece that can move either two squares horizontally and one square vertically or two squares vertically and one square horizontally. A knight’s tour is a sequence of moves of a knight on a chessboard such that the knight visits each square exactly once. In particular, if the knight ends on a square that is one move from the starting square, then the tour is considered closed; otherwise, it is considered open. The earliest record of the closed knight’s tour problem can be traced back to 840 A.D., during which al-Adliar-Rumr and an anonymous chess player provided two distinct solutions for the 8-by-8 knight’s tour [5]. In 1823, Warnsdorff’s (heuristic) rule [9], was developed. This greedy algorithm moves the knight to the square from which the knight will have the fewest subsequent moves. In 1991, Schwenk [7] determined which rectangular boards allow for a closed knight’s tour.

S. Lu Department of Computer Science, University of Oxford, London, UK e-mail: [email protected] C. Yerger (✉) Department of Mathematics and Computer Science, Davidson College, Davidson, NC, USA e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 S. Heuss et al. (eds.), Combinatorics, Graph Theory and Computing, Springer Proceedings in Mathematics & Statistics 462, https://doi.org/10.1007/978-3-031-62166-6_8

111

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Theorem 1 ([7]) An .m×n chessboard with .m ≤ n has a closed knight’s tour unless at least one of the following three conditions holds: 1. m and n are both odd; 2. n .∈ {1, 2, 4}; 3. .m = 3 and .n ∈ {4, 6, 8}. Since Theorem 1 describe the requirements for all rectangular boards, another avenue of research investigated the possibilities of knight’s tours on boards of different shapes. In 1994, Seibel considered the knight’s tour problem on the surface of a torus and on a cylinder [8]. Later in 2002, Cairns studied knight’s tours on the surface of a “pillow”, a structure having only a top and bottom face [2]. Each of the four pairs of corresponding sides of these two faces are connected to form a “pillow chessboard”. In 2009, Kumar investigated knight’s tours on the surface of a cube [4]. At the same time, Kamˇcev gave a characterization of generalized knight’s tours on boards that are the interior of high-dimensional boxes [3]. A helpful general resource on knight’s tour problems is a book [10] by Watkins. Inspired by the results of Kumar and Kamˇcev, we will investigate the existence of closed knight’s tours on the surface of an arbitrarily large box. It turns out this problem had been considered in 2006 by Qing and Watkins [6]. Relying on Seibel’s [8] results, they considered a box as consisting of a single cylindrical chessboard together with a rectangular chessboard at the top and another rectangular chessboard at the bottom. By combining together a tour on the top face, a tour on the bottom face and a tour on the cylindrical part of the box, they can construct a closed tour [6]; however, this algorithm fails when the dimensions of the top face and the lower face are small. Moreover, Qing and Watkins did not go into detail to cover all special cases, nor did they describe their algorithm exhaustively. We believe that a complete proof with full details would be of interest to readers.

2 Preliminaries 2.1 Definitions For a rectangular box with length m, width n, and height r, by treating the squares on its surface as vertices and the knight’s legal moves between any two squares as edges, we can represent the board as a graph. For simplicity, we denote such a chessboard as .G(m, n, r). Throughout this chapter, for a 2-dimensional rectangular board, we simply write .G(m, n), where m represents the number of columns (vertical) and n represents the number of rows (horizontal). We denote the square in column i and row j as .(i, j ), with the top-left corner square being .(1, 1). Moreover, we often divide the squares from a chessboard into different sets. These sets are called groups. When we refer to squares in different groups of a board, we place a prefix before the coordinates

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of the square. For instance, .(1, 1) of group A is written as .A(1, 1). Also, denote an edge from square .(a, b) to square .(c, d) as .(a, b) − (c, d). We will define a move as .(a, b), meaning the knight moves a squares to the right and b squares downwards, where .a, b ∈ {±1, ±2} and .|a| + |b| = 3. To avoid confusion, notice that moves will always be denoted with plus or minus signs. We often consider cylindrical boards and will follow the notation used by Seibel [8]. The dimensions of a cylindrical board are defined similarly to that of a rectangular board, in which .C(m, n) represents a cylinder that is m squares long, which means it has m columns, and n squares wide, which means it has n rows; however, unlike the rectangular situation, a .C(m, n) is different from a .C(n, m). On a standard chessboard, a knight can move to one of at most eight possible squares with a single move. However, based on the definition used by Kumar [4], on the surface of a rectangular solid, a knight may now move to one of at most 10 possible squares with a single move. This is because by unfolding the surface of a rectangular solid along different edges, different configurations will be obtained in which different knight’s moves are possible. Throughout our proof, we will use concatenation of cycles, a common technique used to build a longer cycle from given ones. One explicit example of concatenation is as follows: Let edge .(a, b) − (c, d) be in cycle E and edge .(A, B) − (C, D) be in cycle F which does not intersect with cycle E. Moreover, .(a, b) and .(A, B) are adjacent and .(c, d) and .(C, D) are adjacent. By removing edges .(a, b) − (c, d) and .(A, B)−(C, D) and inserting .(a, b)−(A, B) and .(c, d)−(C, D), the two cycles are concatenated. Hence, cycles E and F are concatenated using edges .(a, b) − (c, d) and .(A, B) − (C, D). Such a concatenation is called .[(a, b) − (A, B)] − [(c, d) − (C, D)] concatenation. Later in this article, we also describe concatenations of tours of rectangles. We will also use insertion, a technique used to include one or several vertices into a cycle to form a longer cycle. For instance, in the case of inserting one vertex, for a cycle with edge .(A, B) − (C, D), if .(a, b) is adjacent to both .(A, B) and .(C, D), an .(A, B) − (a, b) − (C, D) insertion can be applied.

2.2 Theoretical Preliminaries Rectangular Chessboards By comparing two ways of coloring the rectangular chessboard, Schwenk [7] proves the non-existence of a knight’s tour for chessboards of certain dimensions. To complete his proof, he uses construction and induction to show it is feasible to construct knight’s tours on all other boards. In the base case of this induction, Schwenk presents several knight’s tours on boards of various sizes. In the inductive step, Schwenk places a .G(n, 4) along the side of a .G(n, m) and uses special techniques to concatenate the knight’s tours on these two boards to construct a closed knight’s tour on a bigger board. While proving this, Schwenk reveals a special property for .G(n, 4) which will be frequently referred to in our proof.

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Fig. 1 4 × n arrays showing cycles in the proof of Lemma 1. Cycles with solid edges are part of base cases and the inductive steps are shown using the dotted edges and thicker solid edges

Schwenk uses a construction to show that when .n ≥ 5 is odd, any .G(n, 4) can admit two 2n-cycles. Moreover, when .n ≥ 6 is even, any .G(n, 4) can admit four n-cycles. We apply this construction (via a modification of an argument of Schwenk [7]) to help us prove the following lemma: Lemma 1 In G(n, 4), where n ≥ 5 is odd, two 2n-cycles can cover the board, and edges (1, 1) − (3, 2) and (2, 1) − (4, 2) are in different cycles. When n ≥ 6 is even, four n-cycles can cover the board, and edges (1, 1) − (3, 2), (2, 1) − (4, 2), (3, 1) − (5, 2) and (4, 1) − (6, 2) are in different cycles. Proof Using induction, Schwenk has shown that for n ≥ 5, G(n, 4) admits a pair of 2n-cycles when n is odd and four n-cycles when n is even [7]. One of the two cycles in the G(5, 4) base case is shown using solid edges (including the thicker edges) in Fig. 1. Two of the four cycles in the G(6, 4) base case are shown using solid edges as well. In both cases, the inductive steps are shown using the dotted edges and the thicker edges. The other one or two cycles can be obtained by using a reflection of the displayed cycle or cycles about the horizontal axis of the board. As shown in the base case, when n = 5, edges (1, 1)−(3, 2) and (2, 1)−(4, 2) are in different cycles. Hence, the lemma is true for the base case of n = 5. During the induction, the two edges are not used and after induction, they are still in different cycles. Hence, the lemma is true by induction for odd n. By a similar argument, the lemma is true for even n ≥ 6. ⨆ ⨅ Tours on the Cylinder and the Torus Seibel [8] proposes an algorithm that can construct a closed knight’s tour on any cylinder with odd width greater than or equal to 3. We will extend this conclusion to cover all even widths greater than or equal to 6 in the following lemma. Lemma 2 For any cylinder with width of 3 or greater than or equal to 5, a closed knight’s tour can be constructed on the surface. Proof First, Seibel [8] proves that this algorithm works for any cylinder with odd width m ≥ 3. Assume we start from the mth row. Let m = 2k + 1, where k is an integer. The knight takes k (±1, −2) moves to reach the first row. After that, the

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115

Fig. 2 An example of an iteration as described in the proof of Lemma 2

knight takes one (±2, +1) move before taking (k − 1) (±1, +2) moves to reach the (m − 1)st row. The knight can then take one (±2, +1) move to the mth row. This completes one iteration, which means a sequence of moves that cover exactly one square in each row (ignoring the initial position) (Fig. 2). Let us consider the horizontal movement only. Notice that in each iteration, there are 2 two-square horizontal moves and (2k − 1) one-square horizontal moves. Since for every move, the knight is free to move right or left, there are several combinations of moves such that after returning to the mth row, the knight is one square to the right of its initial position. One such combination is when the knight takes the first twosquare horizontal move to the left and the other to the right, and (k − 1) one-square horizontal moves to the left and k one-square horizontal moves to the right. After returning to the mth row, the knight can continue the same iteration and eventually cover the cylindrical surface. As a result, the algorithm works for all cylinders with odd width. We extend this algorithm to cover all cylinders with even width m ≥ 6. For any even m ≥ 6, we can express m = 3 + k, where k is an odd integer greater than or equal 3. We can split the cylinder into two cylinders, with widths of 3 and k respectively. By the previous part of the proof, closed knight’s tours exist in both cylinders. Assume the cylinder with width 3 is at the top. Due to the cyclical symmetry of the cylinder, in all iterations in the top cylinder, the moves from the second row to the third row can be (+2, +1) or (−2, +1). This is also true for the moves connecting the first two rows of the bottom cylinder. Without loss of generality, we can assume all are (+2, +1). Therefore, if moves from appropriate iterations are chosen such that they are exactly one legal move away from each other, the two edges can be used to concatenate the knight’s tours on the two cylinders to construct a complete one. ⨆ ⨅ We now define the concept of superimposibility. If by translating a sequence of moves horizontally on the surface of the cylindrical part of the box (meaning that after the translation, the sequence of moves are identical), it can overlap with another sequence of moves, these two sequences are said to be superimposible. In later proofs, this may be used vertically as well. We now prove the following useful lemma.

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Lemma 3 For a closed tour on a cylinder constructed using Lemma 2, the moves connecting the first row and the second row (and by symmetry, the last row and the second to last row) are superimposible. Proof When the width of the cylinder is odd, since all iterations are identical by construction, they are superimposible. Therefore, the moves mentioned in Lemma 3 are superimposible. When the width of the cylinder is even, we can consider the two composite cylinders with odd widths. Since the moves mentioned in the lemma are not used when concatenating the knight’s tours on the cylinder in our proof for Lemma 2, each move is superimposible with all other moves. Therefore, the lemma is true. ⨆ ⨅

2.3 Knight’s Tours on Boards with Odd Dimensions Although according to Theorem 1, closed tours on rectangular boards both of whose dimensions are odd are impossible, Bi, Butler, DeGraaf and Doebel [1] prove it is possible to delete one specific square so that the remaining board admits a closed knight’s tour. In our proof, we will use the following conclusions from their paper. Theorem 2 For .G(5, 5), when .(i, j ) ∈ {(1, 1), (1, 5), (5, 1), (5, 5)} is removed from the board, the remaining board admits a closed tour. For .G(m, n), where .m ≥ 5 and .n ≥ 7 are both odd, let .1 ≤ i ≤ m and .1 ≤ j ≤ n. When .i + j is even, when .(i, j ) is removed from the board, the remaining part of the board admits a closed tour.

3 Proof of Main Theorem We will now state our main theorem. Theorem 3 There exists a closed knight’s tour on the surface of a rectangular box of any size. In our analysis, we will vary the length of the shortest side length m. For every G(m, n, r), we may assume .m ≤ n and .m ≤ r.

.

3.1 When m ≥ 5 and at Least One Element of {m, n, r} Is Even Without loss of generality, assume edge m is even. We will use m as one of the side lengths of the top and bottom faces. Throughout this paper, the other four

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faces surrounding the top and bottom faces will be named as the cylindrical part or cylinder, unless otherwise specified. Notice .n ≥ 5 and .r ≥ 5. When .m ≥ 5 is even, by Theorem 1, closed tours must exist in the top and bottom faces. Since the height of the cylindrical part is either .n ≥ 5 or .r ≥ 5, we can then use Seibel’s algorithm from Lemma 2 to construct a closed tour on the cylindrical part. Next, consider the concatenation of the bottom face and the cylinder. Since square .(1, 1) of the bottom face is incident with only two edges, namely .(1, 1) − (2, 3) and .(1, 1) − (3, 2), these two edges must be included in the closed tour in the bottom face. Without loss of generality, assume the moves connecting the bottom two rows of the cylinder are all .(+2, +1). Therefore, using one appropriate .(+2, +1) move and .(1, 1) − (3, 2), the two parts can be concatenated. The top face and the cylinder can be concatenated similarly to complete the construction.

3.2 When m ≥ 5 and Three Side Lengths Are Odd Without loss of generality, use .G(m, n) as the top and bottom faces. Since odd number .r ≥ 5 is the width of the cylinder, closed tours must exist in the cylinder. Therefore, we need to concatenate the two .G(m, n) into the cylinder. Let us consider the top face shown in Fig. 3. By Lemma 2, if .(1, 1) is removed, the remaining admits closed tours. Without loss of generality, assume the moves in the top two rows of the cylinder are .(+2, +1). Since .(m, 1) of the top face only has two edges, edge .(m, 1) − (m − 1, 3) must exist in the closed tour on the top face. Therefore, we can use a .[b1 − B1] − [b2 − B2] concatenation. We can use an .A1 − a − A2 insertion to include .(1, 1) of the top face in the closed tour as well. The bottom face can be inserted and concatenated similarly (the existence of these insertions follows by applying Lemma 3) so that a closed tour is constructed.

Fig. 3 One .G(m, n) top face and the two rows next to it (Theorem 3, Sect. 3.2)

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3.3 When m = 4 Case 1: .m = 4 and n Is Odd In this section, use the two .G(4, n) as the top and bottom faces. Consider the concatenation of the top face and the cylindrical part. By Lemma 1, edges .(1, 1) − (2, 3) and .(1, 2) − (2, 4) exist in the two cycles in the top face. Without loss of generality, assume the moves connecting the top two rows of the cylinder are .(−2, +1). Hence, two appropriate .(−2, +1) moves and .(1, 1) − (2, 3) and .(1, 2) − (2, 4) in the top face can be used for concatenation of the top face and the cylinder. The bottom face and the cylinder can be concatenated similarly so that a complete closed tour is constructed. Case 2: m = 4 and n ≥ 6 Is Even By Lemma 1, four n-cycles can cover the board and edges (1, 1) − (2, 3), (1, 2) − (2, 4), (1, 3) − (2, 5) and (1, 4) − (2, 6) are in different cycles. Therefore, similar to Sect. 3.3, Case 1, these four edges and another four (−2, +1) edges from the top two rows of the cylinder can be used to concatenate the top face and the cylinder. Similarly, we can concatenate the bottom face and the cylinder. Case 3: m = 4, n = 4, r ≥ 5 Use the G(4, 4) as the top and bottom faces. When r ≥ 5, according to Lemma 2, a closed knight’s tour can be constructed using Seibel’s algorithm in Lemma 2. Therefore, our job is to insert the 32 squares from the top and bottom faces into the closed tour. Figure 4 shows the top face and the two surrounding rows from the side faces. As shown, after excluding the four corners of the top face, the remaining 12 squares can admit a closed tour. Observe the five pairs of squares labelled A, B, C, D and E (including E1 and E12). For each pair, there is a legal move from one to the other. Since these legal moves are all (+2, +1) and superimposable, they can fit into the closed tour constructed using the algorithm in Lemma 2. Therefore, we can use an [e1 − E1] − [e12 − E12] concatenation and A − a − A, B − b − B, C − c − C and D − d − D insertions to include the top face in the closed tour. Due to the horizontal symmetry of the rectangular box, follow the same procedure for the bottom face to complete the construction of a closed knight’s tour. Explicit Construction for G(4, 4, 4) Kumar provided several closed knight’s tours on G(4, 4, 4) [4]. We will use one of the knight’s tours that he provides. Please refer to the Appendix for this explicit construction. Fig. 4 The G(4, 4) top face and the surrounding two rows (Theorem 3, Sect. 3.3, Case 3)

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Fig. 5 One G(4, 3) and the two rows next to it (Theorem 3, Sect. 3.4, Case 1)

Fig. 6 One G(1, 3) and the two rows next to it (Theorem 3, Sect. 3.4, Case 2)

3.4 When m = 3 Case 1: .n ≡ 0 (mod 4) and .r /= 4 Use .G(n, 3) as the top and bottom faces and concatenate them with the cylindrical part. Since .r ≥ 3 and .r /= 4, the cylinder admits a closed tour using the algorithm in Lemma 2. Since .n ≡ 0 (mod 4), we can divide the top and bottom faces into multiple .G(4, 3). As shown in Fig. 5, each .G(4, 3) admits two cycles. Notice that .A1 − A2 and .B1 − B2 in the cylindrical part are superimposible. Therefore, they can co-exist in the knight’s tour on the cylinder constructed in Lemma 2. By using these two edges and .b1 − b2 and .a1 − a2, we can concatenate one .G(4, 3) with the cylindrical part. Similarly, we can concatenate other .G(4, 3) groups from the top and bottom faces into the cylinder to obtain a closed knight’s tour. Case 2: n ≡ 1 or 2 (mod 4) and r /= 4 Use the G(n, 3) as the top and bottom faces and concatenate them with the cylindrical part, which admits a closed tour that is constructed using the algorithm from Lemma 2. Let us consider the top face first. Starting from one end with length 3, we skip one G(1, 3) and divide the remaining top face into G(4, 3). When n ≡ 1 (mod 4), the remaining is divided into several G(4, 3). When n ≡ 2 (mod 4), another G(1, 3) is left at the other end. As shown in Sect. 3.4, Case 1, all G(4, 3) can be concatenated with the cylinder. Moreover, since A1 − A2, B1 − B2 and C1 − C2 are superimposible, as shown in Fig. 5, we can use A1 − a − A2, B1 − b − B2 and C1 − c − C2 insertions to concatenate a G(1, 3) with the cylindrical part. Therefore, when n ≡ 1 or 2 (mod 4) and r /= 4, a complete closed tour is feasible (Fig. 6). Case 3: n ≡ 3 (mod 4), n ≥ 7 and r /= 4 Use the G(n, 3) as the top and bottom faces. Consider the top face first. After removing a G(7, 3) from one end of the top face, the remaining G(n − 7, 3) can be divided into multiple G(4, 3) boards. By the arguments in Sect. 3.4, Case 1, each G(4, 3) can be concatenated with the

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Fig. 7 One G(7, 3) and the two rows next to it (Theorem 3, Sect. 3.4, Case 3)

cylinder. Therefore, if we can concatenate the G(7, 3) with the cylinder, a closed tour is constructed. As shown in Fig. 7, there is an open tour on G(7, 3), as indicated by the numbers. Moreover, squares A and B, which are adjacent to each other, are adjacent to squares 1 and 21 respectively. Observe that we are now also using numbers to label squares in this and some future tours. Therefore, we may use an [A − 1] − [B − 21] concatenation. Similarly, the bottom face can be concatenated with the cylinder so that a closed tour can be constructed. Case 4: n = 3 and r /= 4 When n = 3, use G(r, 3) as the top and bottom faces instead of the G(n, 3) we used in Sect. 3.4 Cases 1, 2, and 3. Since the width of the cylindrical part is now n = 3, a closed tour on the cylinder must exist according to Lemma 2. When r /= 3, a closed tour can always be constructed based on results from Sect. 3.4, Cases 1, 2, and 3. We will present an explicit construction for G(3, 3, 3) in the Appendix. Case 5: r = 4 The G(r, 3) can be used as the top and bottom faces instead of the G(n, 3) we used in Sect. 3.4, Case 1. Based on results from Sect. 3.4, Case 1, a closed tour is possible when n ≥ 3 and n /= 4. When n = 4, this is G(3, 4, 4), which admits a closed tour constructed using the algorithm in Sect. 3.3, Case 3.

3.5 When m = 2 Case 1: .n ≥ 5 and .r ≥ 5, .r /= 6 Consider the two .G(n, 2) as the top and bottom faces. Due to the limited degrees of freedom on the top and bottom faces, each of them includes one adjacent row from the cylindrical part so that both become .G(n + 2, 4) with the four corners missing. Notice that after removing two rows, the cylinder has a width of .r − 2, which, according to Lemma 2, admits a closed tour. First, ignore the four extra squares (including squares a and b on one side of the .G(n, 4) and the other two on the other side) of the top face so that we have a .G(n, 4). According to the algorithms in Sect. 3.3, Case 1, and 3.3, Case 2, the top face can be concatenated with the cylindrical part. Moreover, as shown in Fig. 8, squares a and b can be inserted into the tour using .A1 − a − A2 and .B1 − b − B2. The other two can be inserted similarly. Applying the same algorithm to the bottom face will produce a complete knight’s tour.

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Fig. 8 Part of the extended top face (Theorem 3, Sect. 3.5, Case 1)

Fig. 9 The .G(4, 2) top face and the two rows next to it (Theorem 3, Sect. 3.5, Case 2)

Fig. 10 The .G(2, 2) top face and the two rows next to it (Theorem 3, Sect. 3.5, Case 2)

Fig. 11 The G(3, 2) top face and adjacent two rows (Theorem 3, Sect. 3.5, Case 2)

Case 2: n = 2, 3, 4 and r ≥ 3 and r /= 4 Use the two G(n, 2) as the top and bottom faces. Since the width of the cylinder is r ≥ 3 and r /= 4, the cylinder admits a closed tour. Now we need to concatenate the 16 squares from the two G(2, 4) with the cylinder. When n = 4, Fig. 9 shows the top face and the unfolded first and second rows of the cylinder. As shown, six superimposible moves are labelled A, B, C, D, E and F. Therefore, we can apply [A1−a1]−[a2−A2], [B1−b1]−[b2−B2], C1−c −C2, D1 − d − D2, E1 − e − E2 and F 1 − f − F 2 insertions to include the top face in the closed tour. Similarly, the bottom face is included in order to complete the knight’s tour. When n = 2, as shown in Fig. 10, we can apply a similar algorithm and use A1 − a − A2, B1 − b − B2, C1 − c − C2 and D1 − d − D2 insertions to include the top face in the knight’s tour. The bottom face can be included in the same way to complete the construction. When n = 3, as shown in Fig. 11, we can apply a similar algorithm and use A1 − a1 − a2 − A2, B1 − b − B2, C1 − c − C2, D1 − d − D2 and E1 − e − E2 insertions (recall that insertions can include more than one square) to include the top face in the knight’s tour. The bottom face can be included in the same way to complete the construction.

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Explicit Constructions Notice that n and r can be swapped in Sect. 3.5, Case 1. Therefore, when n, r ≥ 5, the only case not covered is G(2, 6, 6). Using similar arguments, it is obvious that G(2, 2, 2), G(2, 2, 4) and G(2, 4, 4) are the only cases that are not covered in Sect. 3.5, Case 2. We will use explicit constructions to cover these cases in the Appendix.

3.6 When m = 1 Case 1: .n ≡ 0, 1, 2 (mod 4) and .r ≥ 5 or .r /= 6 Let us consider the .G(n, 1) as the top and bottom faces. First, consider cases when .n ≡ 0 (mod 4). As shown in Fig. 12 (left), we have divided the surface of a .G(1, n, r) into several parts accordingly. Let us first consider how groups A and C can be connected using the concatenation technique. Note group A is a .G(n, 3), groups B are two .G(1, 1) and group C is a .C(2n + 2, r − 2). Since .r ≥ 7 or .r = 5, by Lemma 2, group C admits closed tours. As shown in Sect. 3.4, Case 1, when .n ≡ 0 (mod 4), groups A and C can be connected via concatenation. As shown in Fig. 13 (left), each square is labelled with a letter associated to its group. In this case, a .C1 − B − C2 insertion can be used to concatenate groups B and C. The bottom part and group C can be connected similarly using concatenation so that a closed tour can be constructed. Figure 12 (right) demonstrates a case when .n ≡ 1 (mod 4). Since group A is a .G(n − 1, 3), where 4 divides .n − 1, groups A and C can be concatenated, as shown in Sect. 3.4, Case 1. As shown in Fig. 13 (right), the four squares labelled with B1, b2, b3 and B4 admit a closed tour in this order. Moreover, C1 and C4, which are adjacent to each other, are adjacent to B1 and B4 respectively. Therefore, a .[C1 − B1] − [C4 − B4] concatenation can be used to connect groups B and C. Fig. 12 The front view of two G(1, n, r) (Theorem 3, Sect. 3.6, Case 1)

Fig. 13 The connections of groups B and C via concatenation using a side view of two G(1, n, r) (Theorem 3, Sect. 3.6, Case 1)

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When .n ≡ 2 (mod 4), there are two groups of four squares labelled with B at the two ends of each group A. These groups labeled with B can be concatenated with group C using the same algorithm described in Fig. 13 (right). Therefore, closed tours are feasible for .n ≡ 2 (mod 4) as well. Case 2: n ≡ 3 (mod 4) and n ≥ 7 and r ≥ 5 and r /= 6 Follow the algorithm in Sect. 3.6, Case 1, and divide the surface as shown in Fig. 12 (left). Since n ≡ 3 (mod 4) and n ≥ 7, we can divide group A into a G(7, 3) and several G(4, 3). According to Sect. 3.4, Case 1, and Sect. 3.6, Case 1, G(4, 3) and group B can be concatenated with group C. Hence, we need to concatenate G(7, 3) with group C. Assume G(7, 3) is at one end the group A as shown in Fig. 12 (left). As shown in Fig. 14, the 21 squares labelled with numbers indicate one closed tour in the G(7, 3). Notice squares 1 and 21 are adjacent to each other, making the G(7, 3) admit closed tours. Moreover, without loss of generality, assume the edges in set P of group C is (+2, +1). By using an [A − 5] − [B − 6] concatenation, the G(3, 7) is concatenated with the cylinder. Hence, the top face is concatenated with group C. The bottom part can be concatenated similarly to complete the construction. Case 3: r ∈ {1, 2, 3, 4} and n ≥ 5 or n = 3 Use G(1, r) as the top and bottom faces. Consider the concatenation between the top face and the cylinder first. As shown in Fig. 15, the squares a and b are part of the top face. We define outermost squares, such as square a, as primary squares and the other squares from the top face, such as square b, as secondary squares. Therefore, the top face has at most two secondary squares. For each primary square, we can use an A1 − a − A2 insertion. For each secondary square, we can use a B1−b−B2 insertion. Notice that the insertions of primary squares and secondary squares with the cylinder will not interfere each other. Hence, through such an algorithm, both top and bottom faces can be concatenated with the cylinder and a closed knight’s tour is constructed. Remaining Cases In the previous two sections, cases when r = 6 are left out. However, notice that n and r can be swapped with each other such that cases other Fig. 14 G(7, 3) and parts of the three rows next to it (Theorem 3, Sect. 3.6, Case 2)

Fig. 15 The primary and secondary squares and the two rows next to them (Theorem 3, Sect. 3.6, Case 3)

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than G(1, 6, 6) always fall into Case 1 or 2 of Sect. 3.6. We will provide an explicit construction for G(1, 6, 6) in the Appendix. Other cases that are missed out are when r ∈ {1, 2, 3, 4} and n ∈ {1, 2, 4}. Again, notice n and r can be swapped. Therefore, cases when r = 3 are equivalent with cases when n = 3, which are covered in Sect. 3.6, Case 3. For the remaining cases, including G(1, 1, 1), G(1, 1, 2), G(1, 1, 4), G(1, 2, 2), G(1, 2, 4) and G(1, 4, 4), we will provide explicit constructions in the Appendix.

4 Open Problems 4.1 Generalized Knight An .(a, b)-knight is a generalized knight which can move a units horizontally and b units vertically or b units horizontally and a units vertically during each move. For a .(0, k)-knight, closed knight’s tours are impossible on a 2-dimensional board. However, .G(2, 2, 2) admits a closed knight’s tour by both a .(0, 2)-knight and a .(0, 3)-knight. The explicit constructions for these two cases can be found in the Appendix. It may be interesting to investigate the existence of a closed tour on the surface of a box with generalized knights, including .(0, k)-knights.

4.2 Generalized Board Just as the closed knight’s tour problem was generalized from a 2-dimensional board to a higher-dimensional board, it may be interesting to investigate the possibility of generalizing our problem from the surface of a 3-dimensional box to the surface of a higher-dimensional box.

Appendix In this appendix, we give the explicit constructions discussed earlier in the paper. There are many standard ways to represent the flattened surface of a box and we employ many of these standard ways to show tours on specific box sizes to optimize space on each page (Figs. 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, and 30).

The Existence of a Knight’s Tour on the Surface of Rectangular Boxes Fig. 16 An example on (Theorem 3, Sect. 3.3, explicit construction)

.G(4, 4, 4)

Fig. 17 An example on (Theorem 3, Sect. 3.5, Case 3)

.G(2, 4, 4)

Fig. 18 A closed tour on .G(3, 3, 3) (Theorem 3, Sect. 3.4, Case 4)

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126 Fig. 19 An example on (Theorem 3, Sect. 3.5, Case 3)

.G(2, 2, 2)

Fig. 20 An example on (Theorem 3, Sect. 3.5, Case 3)

.G(2, 2, 4)

Fig. 21 An example on (Theorem 3, Sect. 3.5, Case 3)

.G(2, 6, 6)

Fig. 22 An example on (Theorem 3, Sect. 3.6, remaining cases)

.G(1, 1, 1)

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The Existence of a Knight’s Tour on the Surface of Rectangular Boxes Fig. 23 An example on (Theorem 3, Sect. 3.6, remaining cases)

.G(1, 6, 6)

Fig. 24 An example on (Theorem 3, Sect. 3.6, remaining cases)

.G(1, 1, 2)

Fig. 25 An example on (Theorem 3, Sect. 3.6, remaining cases)

.G(1, 1, 4)

Fig. 26 An example on (Theorem 3, Sect. 3.6, remaining cases)

.G(1, 2, 2)

Fig. 27 An example on (Theorem 3, Sect. 3.6, remaining cases)

.G(1, 2, 4)

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Fig. 28 An example on (Theorem 3, Sect. 3.6, remaining cases)

.G(1, 4, 4)

Fig. 29 An example of on .G(2, 2, 2) (Sect. 4.1)

.(0, 2)-knight

Fig. 30 An example of on .G(2, 2, 2) (Sect. 4.1)

.(0, 3)-knight

References 1. B. Bi, S. Butler, S. DeGraaf and E. Doebel, Knight’s Tours on Boards with Odd Dimensions, Involve, 8 (2015), 615–627. 2. G. Cairns, Pillow Chess, Mathematics Magazine, 70 (2002), 173–186. 3. N. Kamˇcev, Generalized Knight’s Tours, arXiv.org, 2014, available at https://arxiv.org/pdf/ 1311.4109.pdf 4. A. Kumar, A Study of Knight’s Tours on the Surface of a Cube, Crux Mathematicorum with Mathematical Mayhem, 35 (2000), 313–319. 5. H. J. R. Murray, A History of Chess, Oxford University Press, London (1913). 6. Y. Qing, J. J. Watkins, Knight’s Tours for Cubes and Boxes, Congressus Numerantium, 181 (2006), 41–49. 7. A. Schwenk, Which Rectangular Chessboards Have a Knight’s Tour?, Mathematics Magazine, 64 (1991), 325–332. 8. K. Seibel, The Knight’s Tour on the Cylinder and Torus, REU paper, Oregon State University (1994). 9. H. C. Warnsdorff, Des Rösselsprunges einfachste und allgemeinste Lösung, Schmalkalden (1823). 10. J. J. Watkins, Across the Board: The Mathematics of Chessboard Problems, Princeton University Press (2004).

A New Upper Bound for the Site Percolation Threshold of the Square Lattice John C. Wierman

and Samuel P. Oberly

Abstract The upper bound for the site percolation threshold of the square lattice is reduced from 0.679492 to 0.666894, providing the first improvement since 1995. The bound is obtained by using the substitution method with new computational reductions which make calculations for site models more efficient. The substitution method is applied, comparing the site percolation model on a self-matching lattice to the square lattice site percolation model in a two-stage process. Keywords Site percolation · Square lattice · Percolation threshold · Set partitions · Stochastic ordering

1 Introduction A site percolation model consists of an infinite graph G in which each vertex is retained independently with probability p, .0 < p < 1, and deleted otherwise, to obtain a random subgraph. The site percolation threshold .pcsite (G) is the value of the parameter p above which the random subgraph contains an infinite connected component with probability one and below which all components are finite with probability one. The most well-studied two-dimensional lattice model for which the exact percolation threshold value is not known is the site percolation model on the square (a.k.a. simple quadratic) lattice. This strongly contrasts with bond percolation models, for which the square lattice was the first two-dimensional lattice to be exactly solved [5] in 1980. Even determining relatively accurate bounds for the square lattice site percolation threshold is a challenging problem, on which progress has been extremely slow.

J. C. Wierman (✉) · S. P. Oberly Department of Applied Mathematics and Statistics, Johns Hopkins University, Baltimore, MD, USA e-mail: [email protected]; [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 S. Heuss et al. (eds.), Combinatorics, Graph Theory and Computing, Springer Proceedings in Mathematics & Statistics 462, https://doi.org/10.1007/978-3-031-62166-6_9

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Although percolation models were introduced in the 1950s, the first exact solution for the percolation threshold of a periodic lattice was in 1980 by Kesten [5], who proved that the bond percolation threshold is .1/2. Since the bond percolation threshold is less than or equal to the site percolation threshold, this provided the lower bound .

1 ≤ pcsite (Square). 2

In 1982, Higuchi [4] established the strict inequality .

1 < pcsite (Square). 2

The numerical value of the lower bound was improved by Toth [10] in 1985 to 0.503478 ≤ pcsite (Square),

.

and again by Zuev [15] in 1987 to 0.509535 ≤ pcsite (Square) ≤ 0.681890.

.

Prior to this article, the smallest upper bound was established by Wierman [11] in 1995: pcsite (Square) ≤ 0.679492.

.

The most recent previous improvement in bounds was a substantial increase in the lower bound by van den Berg and Ermakov [2] in 1996: 0.556000 < pcsite (Square).

.

Neither bound has been improved since 1996. Thus, the best bounds prior to this article are 0.556 < pcsite (Square) < 0.679492,

.

Our result is pcsite (Square) < 0.666894,

.

shortening the length of the interval between the bounds by more than 10%. For comparison to the bounds, the consensus of recent simulation estimates in the scientific literature is .pcsite (Square) ≈ 0.592746, so neither the upper or lower bound is sharp.

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The derivation of the new upper bound relies on the use of self-matching lattices and the substitution method, both of which will be discussed briefly in Sect. 2. The innovations employed in their adaptation to site percolation models are described in Sect. 3. Section 4 previews research in progress to further improve the upper bound for the square lattice site percolation threshold.

2 Background 2.1 Matching Lattices and Line Graphs The concept of matching lattices was introduced in 1964 by Sykes and Essam [9]. Consider a planar lattice L and a set F of its non-triangular faces. Inserting any diagonal edges in a face necessary to form the complete graph on the vertices of the face is referred to as “close-packing” the face. Thus, a triangular face is considered to be close-packed. Construct a lattice M from L by inserting all possible diagonal edges in each non-triangular face in F . Construct the matching lattice of M, denoted by .M ∗ , from L by close-packing all non-triangular faces of L that are not in F . Note that the matching lattice of .M ∗ is M. The main result of Kesten’s monograph [6] is that the site percolation thresholds of a pair of periodic matching lattices sum to one (Fig. 1).

Fig. 1 An illustration of a pair of matching lattices. Top: An induced subgraph of an infinite planar lattice L. Bottom Left: A subgraph (on the same vertex set) of a lattice M constructed from L by close-packing alternate square faces in the second and sixth columns. Bottom Right: A subgraph (on the same vertex set) of the matching lattice .M ∗ constructed by close-packing all square faces that are not close-packed in M

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A bond percolation model on a graph may be converted into an equivalent site percolation model on its line graph, which is known as the “bond-to-site transformation.” The two models are coupled together by declaring each vertex of the line graph to be retained if and only if the corresponding edge in the bond percolation model is retained. Thus, an infinite connected component exists in the site model if and only if one exists in the bond model. A motivation for the concept of matching lattices is that taking the line graphs of a dual pair of lattices produces a pair of matching lattices. Few site percolation models have exact solutions for the percolation threshold. The site percolation threshold of the triangular lattice is one-half since the lattice is self-matching. The kagome and .(3, 122 ) lattices are solved by the bond-to-site transformation, since they are the line graphs of the hexagonal and subdivided hexagonal lattices, respectively. However, the square lattice is not self-matching nor is it a line graph, because it contains one of the nine forbidden induced subgraphs in Beinecke’s theorem [1].

2.2 Substitution Method The new bound is derived using the substitution method, which has been described in detail in the research monograph by Bollobás and Riordan [3, §6.1] or one of the articles [13, 14] and applied extensively to bond percolation models. It has established the best current bounds for several bond percolation models. In particular, it derived upper and lower bounds which agree to three decimal places for the .(3, 122 ) lattice bond percolation model [13] and to two decimal places for the kagome lattice bond percolation model [12]. However, it has been more difficult to obtain results for site percolation models. This article describes some of the difficulties and adaptations to overcome them. The substitution method compares two percolation models using stochastic ordering of probability measures on connections between vertices on the boundary of finite subgraphs called “substitution regions.” For application to site percolation models, both lattices must be decomposed into vertex-disjoint substitution regions so that the randomness in different regions are stochastically independent. For our application to the square lattice, consequences of this condition are that some edges must be subdivided and that a two-stage process must be used. A very brief description of deriving an upper bound using stochastic ordering is as follows: A configuration on a substitution region is a specification of each edge of the subgraph as retained or deleted. Each configuration determines a set partition of the boundary vertices in which boundary vertices in separate blocks are in different connected components of retained vertices. The probability of a partition is the sum of the probabilities of the configurations that produce it. The set of boundary vertex partitions are partially ordered by refinement. A probability measure P is stochastically larger than a probability measure Q if .P [U ] ≥ Q[U ] for every upset U . In our application, .Pp is the probability measure on boundary vertex

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partitions of the unsolved lattice with vertices retained with probability p, and Q is the probability measure on boundary vertex partitions of a reference lattice with vertices retained with probability equal to or larger than its percolation threshold. If .Pp is stochastically larger than Q, then p is an upper bound for the percolation threshold of the unsolved lattice. While the approach seems relatively simple, the computations required increase super-exponentially as a function of the number of boundary vertices. Major reductions of computations have been accomplished using techniques involving graph-welding, symmetry, non-crossing partitions, and a network flow model [7, 8]. Still, for bond percolation models, it has only been possible to do the calculations symbolically for substitution regions with at most ten boundary vertices. In our application to the square lattice site percolation model, we consider substitution regions with 22 boundary vertices, and are able to adapt these techniques to complete the computations. Section 3 focuses on the details of the implementation for site models from that for bond models, with only minor mention of aspects that have become standard in applications to bond models.

3 Application of the Method In our application of the substitution method to the square lattice, the computational techniques involved in finding the partition probability measures, the use of symmetry, and the conversion to a network flow model are similar to those for bond percolation models. Adaptations for the site model that are needed are finding an appropriate solved reference lattice and dealing with a much larger number of boundary vertices. In Sect. 3.1 we describe an uncommon reference lattice designed to serve our purpose. Section 3.2 introduces the two-stage process and the substitution regions for each stage. Adaptations due to the large number of boundary vertices and the relatively small numbers of random vertices are discussed in Sect. 3.3.

3.1 Reference Lattice For the solved reference lattice, we designed a lattice consisting of a square lattice with alternating rows of squares close-packed. See Fig. 2. It is self-matching, so, by Kesten’s theorem, it has site percolation threshold equal to .1/2. A reason for its choice is that can be decomposed into vertex-disjoint substitution regions which have both rotation and reflection symmetry.

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Fig. 2 The reference lattice. Kesten’s result applies to this self-matching lattice, showing that its site percolation threshold is .1/2

3.2 Two Stage Process For the application of the substitution method to a site percolation model, each lattice must be decomposed into vertex-disjoint isomorphic substitution regions. To accomplish this, vertices of a lattice cannot be on the boundary of two substitution regions, so certain edges must be subdivided by inserting a vertex which is considered to be open with probability one, to create the boundary vertices of the substitution region. We need a substitution which transforms the reference lattice into a square lattice by removing the diagonal edges from all close-packed faces. The requirement of vertex-disjoint substitution regions makes this more difficult for site percolation models than for bond percolation models. While we would wish to consider a substitution region that is a 3-by-2 rectangle containing two close-packed squares from each of two rows of close-packed squares, we must enlarge that region by subdividing the edges incident to the 3by-2 region, inserting a boundary vertex when subdividing each edge. See Fig. 3. This has two important effects. There are 22 boundary vertices rather than the 10 vertices on the boundary of the 3-by-2 region. The close-packed squares including subdivided edges do not have their diagonal edges removed in the application of the substitution method with these substitution regions, so a second stage is required.

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Fig. 3 Dashed rectangles indicate the substitution regions for the first stage, in which the diagonals are removed in half of the close-packed faces

The application of the substitution method with the substitution regions in Fig. 3 eliminates the diagonal edges from two-thirds of the close-packed faces, producing the lattice in Fig. 4. Our calculations produce an upper bound for the site percolation threshold of the resulting lattice. The rest of the close-packing is removed in a second application of the substitution method. The lattice produced by the first stage serves as the reference lattice for the second stage, with its upper bound being used as the retention probability parameter for its partition probability measure. Again, expanded 3-by-2 rectangular substitution regions may be used, but because some close-packing was removed in the first stage, the number of boundary vertices is reduced to 14. However, the regions have fewer symmetries. These differences would decrease and increase, respectively, the computational effort required. In this case, it remains possible to complete the calculations and derive the desired upper bound for the square lattice site percolation threshold.

3.3 Boundary Vertex and Partition Issues For site percolation models, one complication is that a vertex in a substitution region may be connected to multiple boundary vertices. Thus, a substitution region in a site

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Fig. 4 The intermediate lattice after the first stage. Dashed rectangles indicate the substitution regions for the second stage, in which the diagonals are deleted from all remaining close-packed faces

model typically has many more boundary vertices than a corresponding region in a bond model. However, we can reduce the number of boundary vertices if there are more than two boundary vertices adjacent to a vertex. If the vertex is retained, then the set of adjacent boundary vertices is included in one block of a partition. If the vertex is not retained, each of the adjacent boundary vertices is in a separate block of the partition. Thus, we may reduce consideration to two of the adjacent boundary vertices, since their relationship in a partition determines whether the vertex is open or closed. Hence, for the purposes of the substitution method calculations, we may reduce the number of boundary vertices adjacent to each vertex to a maximum of two. The substitution method compares probability distributions on the set of partitions of the boundary vertices of the substitution region. For bond percolation models, if both lattices are planar, a substantial computational reduction was achieved by restricting to noncrossing partitions, since all crossing partitions have probability zero in both the problem model and the reference model. For a region with n boundary vertices, the number of set partitions is the n-th Bell number, while the number of noncrossing partitions is the n-th Catalan number. For our substitution regions, the number of boundary vertices that occurs in the first stage

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is 22. Consequently, there are .Bell(22) = 4, 506, 715, 738, 447, 323 partitions and Catalan(22) = 91, 482, 563, 640 noncrossing partitions. Rather than generating all the set partitions or noncrossing partitions of the boundary vertices, then eliminating those that have zero probability, we focus on the much smaller number of configurations, each of which has positive probability. Since there are only 10 vertices in the region which can be retained or deleted, there are only .210 = 1024 configurations. After a dihedral symmetry reduction to equivalence classes of partitions, there are 663 classes. This sufficiently reduces the calculations so they can be completed within reasonable time and memory constraints. The calculations were all done symbolically in MATLAB to obtain exact results, avoiding round-off errors and numerical approximations. The result of the stage 1 calculation is an upper bound of

.

.

100192053 < 0.597192 167772160

for the site percolation threshold of the intermediate lattice. The substitution regions for the second stage have only 14 boundary vertices, but the same number of vertices with randomness, so have 1024 configurations. Since there is less symmetry, the number of classes is only reduced to 895, which is still sufficiently small that the calculations can be completed. The upper bound obtained for the square lattice site percolation threshold is .

111886203 < 0.666894. 167772160

4 Summary and Continuing Research Adaptations to the substitution method allow it to be applied more efficiently to site percolation models, providing the first improvement since 1996 of a bound for the site percolation threshold of the square lattice. The key adaptations are designing a self-matching reference lattice, using a two-stage process to transform the reference lattice into the unsolved lattice, and constructing only the partitions that have positive probability. In current continuing research, we are applying these adaptations to a larger substitution region of the square lattice, using a two-stage process. The comparison will use a 3-by-3 square region to compare the square lattice with the matching lattice of the square lattice. Based on past experience, it is expected that the upper bound for the square lattice site percolation threshold will be an improvement over the result of this article. Future research will apply the adaptations from this article to site percolation models on other lattices commonly studied in percolation theory, such as the lattices corresponding to the Archimedean tilings.

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Acknowledgments The authors gratefully acknowledge funding to support undergraduate research from Johns Hopkins University via the Acheson J. Duncan Fund for the Advancement of Research in Statistics.

References 1. Beinecke, Lowell W. (1968) Derived graphs and digraphs. In Beiträge zur Graphentheorie, Tuebner, 17–33. 2. van den Berg, J. and Ermakov, A. (1996) A new lower bound for the critical probability of site percolation on the square lattice. Random Structures and Algorithms, 8, 199–212. 3. Bollobás, Béla and Riordan, Oliver (2006) Percolation. Cambridge University Press. 4. Higuchi, Y. (1982) Coexistence of infinite (*) clusters: A remark on the square lattice site percolation. Zeitschrift für Wahrscheinlichkeitstheorie und Verwandete Gebeite 61, 75–81. 5. Kesten, Harry (1980) The critical probability of bond percolation on the square lattice is 1/2. Communications in Mathematical Physics 74, 41–59. 6. Kesten, Harry (1982) Percolation Theory for Mathematicians, Birkhäuser, Boston. 7. May, William D. and Wierman, John C. (2005) Using symmetry to improve percolation threshold bounds. Combinatorics, Probability and Computing 14, 549–566. 8. May, William D. and Wierman, John C. 2007 The application of non-crossing partitions to improving percolation threshold bounds. Combinatorics, Probability and Computing 17, 285– 307. 9. Sykes, M. F. and Essam, J. W. (1964) Exact critical probabilities for site and bond problems in two dimensions. Journal of Mathematical Physics 5, 1117–1127. 10. Tóth, Balint (1985) A lower bound for the critical probability of the square lattice percolation. Zeitschrift für Wahrscheinlichkeitstheorie und Verwandete Gebeite 69, 19–22. 11. Wierman, John C. (1995) Substitution method critical probability bounds for the square lattice site percolation model. Combinatorics, Probability, and Computing, 4, 181–188. 12. Wierman John C., Yu Gaoran, and Huang, T. (2015) A disproof of Tsallis’ bond percolation threshold for the kagome lattice. Electronic Journal of Combinatorics 22, P2.52. 13. Wierman, John C. (2016) Tight bounds for the bond percolation threshold of the (3, 122 ) lattice. Journal of Physics A 49, 475002. 14. Wierman, John C. (2017) Strict inequalities between bond percolation thresholds of Archimedean lattices. Congressus Numerantium 229, 231–244. 15. Zuev, S. A. (1987) Percolation threshold bounds for the square lattice. Theory of Probability and its Applications (In Russian) 32, 606–609 (551–553 in translation).

Prime, Composite and Fundamental Kirchhoff Graphs Jessica Wang and Joseph D. Fehribach

Abstract A Kirchhoff graph is a vector graph with orthogonal cycles and vertex cuts. An algorithm has been developed that constructs all the Kirchhoff graphs up to a fixed edge multiplicity. This algorithm is used to explore the structure of prime Kirchhoff graph tilings. The existence of infinitely many prime Kirchhoff graphs for a given set of fundamental Kirchhoff graphs is established, as is the existence of a minimal multiplicity for Kirchhoff graphs to exist. Keywords Kirchhoff graphs · Graph theory · Uniformity families

1 Introduction In recent years, Kirchhoff graphs have been studied extensively by the second author and his colleagues and students (see Fehribach [1, 2], Fehribach and McDonald [3], Reese, Fehribach, Paffenroth and Servatius [4–8]). They are of interest in part because they can serve as circuit diagrams for reaction networks [9]. Previous studies have considered the construction and properties of individual Kirchhoff graphs. The present work considers the structure of families of Kirchhoff graphs all of which are associated with the same set of edge vectors. In order to do this, we have developed a numerical method for constructing all Kirchhoff graphs up to a certain size for a given edge vector set. The results of our computations allow the definition of prime, composite and fundamental Kirchhoff graphs, motivate the proofs of several results, and lead to a number of open questions regarding the structure of these Kirchhoff graph families. A Kirchhoff graph is a vector graph whose edges are vectors (or whose edges are assigned vectors) that satisfy an orthogonality condition between its cycles and its

J. Wang · J. D. Fehribach (✉) Worcester Polytechnic Institute, Worcester, MA, USA e-mail: [email protected]; [email protected] https://www.wpi.edu/~bach © The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 S. Heuss et al. (eds.), Combinatorics, Graph Theory and Computing, Springer Proceedings in Mathematics & Statistics 462, https://doi.org/10.1007/978-3-031-62166-6_10

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vertex cuts. There is a cycle in the graph only when the corresponding vectors add to zero in the vector space. Consider a set .S := {s 1 , s 2 , . . . , s n } of vectors in a vector space .V over .Q. These are the edge vectors for our vector graphs. For simplicity, suppose that no vector in S is a scalar multiple of another, and suppose there is a k where .1 < k < n so that .{s 1 , s 2 , . . . , s k } forms a basis for .Span(S). Then for ' .[s 1 , s 2 , . . . s n ], a row vector of vectors, there is a coefficient matrix .C such that ' ' ' .[s 1 , s 2 , . . . s n ]·[C /−In−k ] = 0 where .[C /−In−k ] is a block matrix with .C over + ' .−In−k . Let .q ∈ Z be the least common multiple of denominators of .C , and define ' .C := qC so the entries of C are integers. Then define .N := [C/−qIn−k ] as the null matrix for S, and .R := [qIk |C] as the row matrix for S. Specifically, the columns of N form a basis for .Null(R), and the columns of R can be used to represent the vectors in S since all finite-dimensional vector spaces of a given dimension k over a given field are isomorphic. This means that any matrix A that is row equivalent to R has the same row space and null space as R, and thus has the same set of Kirchhoff graphs. The orthogonality condition mentioned above corresponds to orthocomplementary of the matrices R and N. For a vertex v in a vector graph .G, the vertex cut of v, denoted .λ(v) = {λ1 , . . . , λn }, has entries corresponding to the vectors .s 1 , . . . , s n . For each i, entry .λi is the net number of times .s i exits vertex v. Add 1 to .λi for each copy of .s i that exits v; subtract 1 from .λi for each copy of .s i that enters v. Thus .λi is zero if .s i is not incident on v, or if .s i enters and exits the same number of times. A cycle C in a vector graph .G is an alternating sequence of vertices and edges that starts and ends with the same vertex in which no vertex appears twice except for the first and the last. Cycles in a vector graph corresponds to linear combinations of the edge vectors .s 1 , . . . , s n that add to the zero vector. The cycle vector for a cycle C, denoted .χ (C) = {χ1 , . . . , χn }, has entries corresponding to vectors .s 1 , . . . , s n . For each i, entry .χi is the net number of times .s i appears in the cycle. Add 1 to the i-th component each time C traverses an .s i in the forward direction, and subtract 1 for each .s i in the backward direction. A Kirchhoff graph for a set of edge vectors S is then vector graph satisfying two conditions: 1. For each vertex v of .G, .λ(v) ∈ Row(R). 2. For each cycle C of .G, .χ (C) ∈ Null(R), and there is a cycle basis for the cycle space of .G corresponding to a basis for .Null(R). This implies that for each vertex and each cycle of .G, .λ(v) ⊥ χ (C). As a simple example of Kirchhoff graphs, consider the matrix ┌ R1 =

.

2 0 1 1 0 2 1 −1

┐

with .S1 = {s 1 , s 2 , s 3 , s 4 } and the columns of .R1 being a representation of the vectors of S. Then two Kirchhoff graphs for .R1 and .S1 are given in Fig. 1. Notice that there are two copies of each of the edge vectors in each of the Kirchhoff graphs in Fig. 1. A Kirchhoff graph is uniform if and only if each of its edge vectors appear the same number of times. In addition, a vector graph is

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Fig. 1 Kirchhoff graphs .F 1 (left) and .F 2 (right) for the edge vectors .s 1 = [2, 0]T , .s 2 = [0, 2]T , = [1, 1]T , .s 4 = [1, −1]T embedded in the Euclidean plane. The small 2 on .s 1 and .s 2 in .F 2 indicate that two copies of these edge vectors lie on top of each other in this Kirchhoff graph. Notice that their vertex cuts lie in the row space, and the cycles form a basis for the null space

.s 3

vector 2-connected if and only if for any pair of vector edges .s i and .s j , there exists a cycle C such that the cycle vector .χ (C) is nonzero with respect to both .s i and .s j . Reese, Fehribach, Paffenroth and Servatius [6, 7] proved the following: Theorem 1 Every vector 2-connected Kirchhoff graph is uniform. One important way that a Kirchhoff graph will fail to be vector 2-connected is if the matrix C has a row of zeros. All Kirchhoff graphs considered here are vector 2-connected and hence uniform. Given that a Kirchhoff graph is vector 2-connected and uniform, let .m = m(G) be the edge multiplicity (or simply the multiplicity) of .G, the number of times each edge vector appears in .G. A second important property of Kirchhoff graphs is chirality: Given a Kirchhoff graph .G embedded in the Euclidean plane, its chiral graph is obtained by rotating .G through 180 degrees about the origin, then reversing each edge vector. While this process does not precisely produce the mirror image (the meaning of the word “chiral” in chemistry), it is faithful to the key idea. Theorem 2 If .G is a Kirchhoff graph, then so is its chiral. A Kirchhoff graph is a self-chiral if and only if its chiral is itself. In other words, it is invariant under the chiral action.

2 Finding Kirchhoff Graphs Using Uniformity For uniform Kirchhoff graphs, their uniformity can be used as a basis for a exhaustive backtracking constructive search algorithm. Let a set of edge vectors S and thus a row matrix .R = [qI |C] be given. First a list is constructed of all the vertex cuts that both lie in .Row(R) and have entries whose absolute values are no greater than a given multiplicity bound .mmax . Then starting from an anchor vertex, consider whether or not the first entry on this list might be the vertex cut for this base vertex. If it could be, one provisionally accepts it and then moves to the next

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vertex and asks the same question; if it cannot be, then one moves to the second entry on the list and checks if it might be the vertex cut for this base vertex. In this way, one checks all of the entries from the list at each vertex. When the final list entry is rejected at a given vertex, one goes back to the previous vertex, discards the current provisional vertex cut, and tries the next entry from the list for this vertex. The process continues until either a uniform Kirchhoff graph is found or all possible vertex cuts are rejected. Using this algorithm, for a given S, one can find all of the uniform Kirchhoff graphs whose multiplicities do not exceed the multiplicity bound .mmax , or it can be shown that no such Kirchhoff graph exists. The brief outline above of our algorithm is expanded with more detail in the next subsection. Our code that implements this algorithm in java can be found at https:// github.com/Jessica-Wang-Math/Kirchhoff.git.

2.1 Structure of the Algorithm What follows is a somewhat more detailed description of the exhaustive backtracking search algorithm for a given edge vector set S (or matrix R) and multiplicity .mmax . 1. Find all possible vertex cuts with entries between .−mmax and .mmax by finding all linear combinations of the row vectors of R. Let .Λ be the list of these vertex cuts in an arbitrary order. Initialize .T as an empty list for us to add potential vertices to as graph construction continues; this serves as our to-do list. 2. Place a starting or anchor vertex at the origin in k dimensional Euclidean space, and add this anchor vertex to our to-do list. Because of the way that R and N are defined, all of the vertices will occur at integral coordinates. In addition, because every Kirchhoff graph is finite, no vertex needs to occur at coordinates whose sum is negative. In other words, no vertex needs to be below and behind (to the left of) the anchor vertex, and our construction can begin with the southwest vertex or edge(s) of the graph. 3. Assign the first vertex cut from .Λ to the anchor vertex, adding in the required edge vectors and required incident vertices. If any of these vertices have coordinates whose sum is negative, delete all of these edge vectors and incident vertices, and consider the next vertex cut from .Λ. Otherwise, add the neighboring incident vertices to the to-do list .T. 4. Go to the next vertex .vi in the graph (according to the order in .T) and check whether it is already in .Row(R). Assuming that it is not, assign the first vertex cut from .Λ to it, and check whether any of the new incident vertices (1) have coordinates whose sum is negative, or (2) result in the edge vector count for any edge vector exceeding .mmax . If either of these occur, delete all of the new edge vectors and incident vertices, and check the next vertex cut from .Λ. If not, add all of the new incident vertices to .T if they are not already on the list. Notice that some of the new incident vertices may have previously been removed from

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T, but the newly added vectors may imply that their vertex cuts are no longer in Row(R). Delete .vi from .T, and go to the next vertex on .T after .vi . 5. When the final vertex cut on .Λ is eliminated at a vertex, that vertex is abandoned, and we move back to the previous vertex which is placed back at the top of the to-do list. For this previous vertex, we consider the next vertex cut on .Λ. The new edges and incident vertices required for this next vertex cut are added, the to-do list is updated, and the process moves back to the previous step (4) in the algorithm. 6. The process ends either when there are no vertices left on the to-do list .T (in which case a Kirchhoff graph is found, and we consider the next vertex cut from .Λ at the anchor vertex), or when the last possible vertex cut on .Λ is eliminated at the anchor vertex (in which case no further Kirchhoff graphs exist, and the entire process ends). Thus the algorithm either finds all Kirchhoff graphs .G with .m(G) ≤ mmax , or it shows that no Kirchhoff graph exists for S with edge multiplicity not exceeding .mmax . . .

2.2 Kirchhoff Graph Examples Found by the Algorithm The algorithm discussed above shows that the two Kirchhoff graphs shown in Fig. 1 are in fact the only ones for .R1 with .mmax = 2. Given ┌ R2 =

.

2 0 1 1 0 2 3 1

┐

and an upper multiplicity bound .mmax = 6, the algorithm finds 16 non-trivial Kirchhoff graphs, as shown in Fig. 2. Notice that .mmax = 6 is the smallest multiplicity for any Kirchhoff graphs associated with .R2 . The first 8 Kirchhoff graphs are self-chirals, while the rest are chiral pairs. For another example, let ┌ R3 =

.

1 0 2 1 0 1 1 2

┐

and let again .mmax = 6. In this case, our algorithm finds 4 prime Kirchhoff graphs, as shown in Fig. 3. Two form a chiral pair, and two are self-chirals.

3 Tiling of Kirchhoff Graphs This section considers the structure of Kirchhoff graph families like those shown above. Doing this requires the concept of tiling, which in turn requires the operations of addition and subtraction.

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Fig. 2 Sixteen prime Kirchhoff graphs for matrix .R2 . The bottom eight are four chiral pairs; the top eight are self-chirals

Fig. 3 Four prime Kirchhoff graphs for matrix .R3 . The bottom two are a chiral pair; the top two are self-chirals

Definition 1 Given two Kirchhoff graphs .G1 , G2 , and a coordinate .x ∈ Zk , define the sum .(G1 + G2 , x) as the union of .G1 with its anchor vertex placed at the origin and .G2 with its anchor vertex placed at the coordinate .x. Then .V (G1 + G2 ) = V (G1 ) ∪ V (G2 ) and .m(G1 + G2 ) = m(G1 ) + m(G2 ). So all copies of the edge vectors from both .G1 and .G2 are present in their sum. When it is unambiguous,

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write .G1 + G2 as a shorthand. Also for simplicity, we only consider sums that preserve vector 2-connectivity. In addition, define the difference .(G1 − G2 , x) as the removal of .G2 from .G1 assuming that there is a copy of .G2 as a subgraph of .G1 anchored at .x. Collectively the process of repeatedly adding and/or subtracting Kirchhoff graphs is called tiling, and the resulting Kirchhoff graph is called a tiling. For the Kirchhoff graphs in Fig. 1, their sum .(F 1 + F 2 , (1, 1)) is shown in Fig. 4. Notice that the sum or difference of Kirchhoff graphs must itself also be a Kirchhoff graph. By tiling Kirchhoff graphs in various ways, one can generate a wide variety of Kirchhoff graphs. Definition 2 Given N Kirchhoff graphs .G1 , . . . , GN , the set of all Kirchhoff graphs that can be constructed by tiling these N graphs is .〈G1 , . . . , GN 〉 := {a1 G1 + . . . + aN GN : ai ∈ Z}, where .ai Gi is any Kirchhoff graph that can be constructed by tiling .Gi , using any anchor coordinate. The .ai can be negative only when there exists .|ai | copies of .Gi in the larger Kirchhoff graph. Now consider the principal definition of this section: Definition 3 A Kirchhoff graph .G is prime if and only if .G has no nontrivial Kirchhoff subgraph decomposition. In other words, .G cannot be written as .G1 + G2 where both .G1 and .G2 are nontrivial Kirchhoff graphs. A Kirchhoff graph that is not prime is composite. The Kirchhoff graphs shown in Figs. 1, 2, and 3 are all prime; the one in Fig. 4 is of course composite. Indeed it might seem that all Kirchhoff graphs that are tilings of smaller Kirchhoff graphs will themselves be composite. Perhaps surprisingly, this is not the case: Consider the two Kirchhoff graphs in Fig. 5. The Kirchhoff graph on the left is a .C 1 := 4F 1 (a tiling by addition of four copies of .F 1 ); it is clearly composite. The Kirchhoff graph on the right is .P 1 := 4F 1 − F 2 (the copy of .F 2 in the middle has been removed); it is prime. This can be seen because if any edge vector is removed, the vertex cuts for incident vertices will no longer lie in .Row(R1 ), and the only way to return all the vertex cuts to .Row(R1 ) is to remove all the edge vectors, meaning that there is no Kirchhoff subgraph. The above example makes clear that composite Kirchhoff graphs may not have unique prime decompositions. In this example, .C 1 = 4F 1 = P 1 + F 2 . On the other hand, there are infinitely many prime Kirchhoff graphs. Fig. 4 An addition example + F 2 , (1, 1)) for the row matrix .R1

.(F 1

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Fig. 5 The Kirchhoff graph on the left is the composite .C 1 := 4F 1 ; the one on the right is the prime .P 1 := 4F 1 − F 2

Theorem 3 For the two prime Kirchhoff graphs .F 1 and .F 2 for the matrix ┌ R1 =

.

2 0 1 1 0 2 1 −1

┐

with .mmax = 2 (see Fig. 1), .〈F 1 , F 2 〉 contains infinitely many prime Kirchhoff graphs. Proof Constructing arbitrarily large prime Kirchhoff graphs that are in .〈F 1 , F 2 〉 is a simple extension of the construction of the prime Kirchhoff graph in Fig. 5. Simply form .6F 1 − 2F 2 by adding two more copies of .F 1 to the prime Kirchhoff graph in Fig. 5, and then by subtracting .F 2 from the middle. Notice that the addition is exactly what is needed to create the copy of .F 2 and thus make possible the subtraction. One can repeat this tiling until the prime Kirchhoff graph of the desired size is achieved. The next two such prime Kirchhoff graphs in .〈F 1 , F 2 〉 are shown in Fig. 6. Remark 1 Although it is hard to describe in general, this sort of construction of prime Kirchhoff graphs of arbitrary size would seem to be possible in .〈K 1 , K 2 〉 for any pair of prime Kirchhoff graphs of minimal multiplicity for a given row matrix R. This is particularly true when the generating Kirchhoff graphs .K 1 and .K 2 are self-chiral.

4 Fundamental Graphs, Tiling Structure One final important implication of this work is a beginning of an understanding of the structure of Kirchhoff graph tilings. Notice that for a given set of edge vectors S

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Fig. 6 Two larger prime Kirchhoff graphs generated by tiling for .R1

and the corresponding row matrix R, there is a minimum edge multiplicity number m∗ below which there can be no Kirchhoff graphs. With the assumptions that no edge vector is a constant multiple of another, the smallest possible Kirchhoff graph is a vector triangle having .n = 3, .k = 2 and .m∗ = 1. For .m = m∗ there will be a finite number of Kirchhoff graphs, and each of these will be prime since any proper subgraph would have fewer than .m∗ edge vectors. It is frequently the case, however, that one or more of these graphs is in fact a tiling of others. An example of this is shown in Fig. 3 for .R3 where any one of the four Kirchhoff graphs is a tiling of the other three. This structure leads to one additional definition, that of fundamental Kirchhoff graphs.

.

Definition 4 A fundamental set for R and S is a minimal generating set in terms of tiling with respect to both multiplicity and cardinality. Members of a fundamental set are called fundamental Kirchhoff graphs. For .R1 , both of the Kirchhoff graphs in Fig. 1 are fundamental. For .R3 , any three of the Kirchhoff graphs in Fig. 3 are fundamental. For .R2 , no more than twelve of the sixteen are fundamental. Based on our current computations, all known larger Kirchhoff graphs are tilings of the graphs in the fundamental set, and thus all have an edge vector multiplicity that is an integral multiple of .m∗ , though this final point remains unproven.

5 Open Questions The work completed so far on prime, composite and fundamental Kirchhoff graphs has led to a number of open questions: – Given a matrix R, is it possible to predict .m∗ , the smallest m such that nontrivial Kirchhoff graphs exist, without constructing the graphs?

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– Given a matrix R and minimal multiplicity .m∗ , how many prime Kirchhoff graphs are there with .m = m∗ ? – Is there a matrix R with fundamental set .{F 1 , . . . , F 𝓁 }, but where there is also a larger Kirchhoff graph .F + /∈ 〈F 1 , . . . , F 𝓁 〉? – Is there a condition on R for the existence of chiral pairs? Acknowledgments The authors wish to thank Padraig Ó Catháin, Randy Paffenroth and Brigitte Servatius for many helpful discussions of this work.

References 1. J.D. Fehribach, Vector-space methods and Kirchhoff graphs for reaction networks, SIAM Journal of Applied Mathematics 70 (2009) 543–562. 2. J.D. Fehribach, Matrices and their Kirchhoff graphs, Ars Mathematica Contemporanea 9 (2015) 125–144. 3. J. D. Fehribach, J. J. McDonald, Matrices and Kirchhoff graphs, a rank-two, nullity-two construction, Congressus Numerantium 230 (2018) 199–207. 4. T. Reese, J.D. Fehribach & R. Paffenroth, Duality in geometric graphs: vector graphs, Kirchhoff graphs and Maxwell reciprocal figures, Symmetry 8 (2016) 1–28. 5. T. Reese, J.D. Fehribach, R. Paffenroth & B. Servatius, Matrices over finite fields and their Kirchhoff graphs, Linear Algebra and its Applications 547 (2018) 128–147. 6. T. Reese, J.D. Fehribach, R. Paffenroth & B. Servatius, Uniform Kirchhoff graphs, Linear Algebra and its Applications 566 (2019) 1–16. 7. J. D. Fehribach, Kirchhoff graph uniformity, Congressus Numerantium 233 (2019) 143–150. 8. T. Reese, J.D. Fehribach & R. Paffenroth, Equitable edge partitions and Kirchhoff graphs, Linear Algebra and its Applications 639 (2022) 225–242. 9. J. D. Fehribach, Kirchhoff graphs and stoichiometry, Journal of the Electrochemical Society 170 (2023) 083504.

On the Minimum Locating Number of Graphs with a Given Order Sul-young Choi and Puhua Guan

Abstract A locating set S in a connected graph is a set of vertices satisfying that N(u) ∩ S is unique for each vertex u not in S. A locating set can be considered as a set of sensors which can determine the exact location of an intruder. The size of a smallest locating set of a graph G is called the locating number of the graph and denoted by .ln(G). We show that .min {ln(G) : G is a connected graph with n vertices .} = s when .2s−1 + (s − 1) < n ≤ 2s + s.

.

Keywords Locating set · Dominating set

1 Introduction Let .G = (V (G), E(G)) be a simple connected graph. For a vertex u in G, the neighborhood of u, .N (u), is the set of vertices adjacent to u. A subset S of .V (G) is called a locating set when the intersection of S with the neighborhood of each vertex u in .V (G) \ S is unique, i.e., .N (u) ∩ S /= N (v) ∩ S for any two vertices u and v in .V (G) \ S. The size of a smallest locating set of a graph G is called the locating number of G and denoted by .ln(G). A locating set can be considered as a set of monitors or sensors which can determine the exact location of an intruder when we assume that each device can detect an intruder at one of its neighboring vertices. However, these devices are not able to differentiate an intruder at a vertex x in .S with an intruder at a vertex y in .V (G) \ S satisfying .N (y) ∩ S = {x} if we assume each device can also detect an intruder at the vertex that it is installed. Locating sets were introduced by Slater in [2] and their definition is not same as ours in this paper.

S.-y. Choi (✉) Department of Mathematics, Statistics and Actuarial Science, Le Moyne College, Syracuse, NY, USA e-mail: [email protected] P. Guan Department of Mathematics, University of Puerto Rico-Rio Piedras, Rio Piedras, Puerto Rico © The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 S. Heuss et al. (eds.), Combinatorics, Graph Theory and Computing, Springer Proceedings in Mathematics & Statistics 462, https://doi.org/10.1007/978-3-031-62166-6_11

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A locating set in [2] is defined as an ordered set of vertices such that a vertex u has a unique sequence of distances between u and each vertex in the locating set, which also can be utilized as a means to find the exact location of an intruder. The concept of locating sets has been evolved and merged with that of a dominating set so that it can model real-world situations more closely, such as a locating dominating set in [3] and an open neighborhood locating dominating set in [4]. In this chapter we are interested in the locating sets that Omega and Canoy studied in [1]. Omega and Canoy showed locating numbers for complete graphs and graphs with small order (.n ≤ 6) while determining relationships between the locating numbers and the strictly locating numbers. Here a set of vertices S of a graph G is a strictly locating set if S is locating and .N (u) ∩ S /= S for all u in .V (G) \ S. The strictly locating number .sln(G) of a graph G is similarly defined. In addition they characterized the locating sets in the join and corona of graphs. We will look into the locating numbers of cycles and paths in Sect. 2, and the lower bound of locating numbers of graphs with a given order in Sect. 3.

2 Simple Results It is obvious that a graph G with n vertices satisfies .1 ≤ ln(G) ≤ n − 1. We will assume a graph has at least four vertices and a locating set at least two vertices. For a graph with a locating set, we will label a vertex with 1 if the vertex belongs to the locating set; and label with 0, otherwise. As shown in Fig. 1, the locating number of a graph is not necessarily larger than or equal to that of its subgraph. Let us consider the locating number of a cycle .Cn with n vertices. Suppose S is a locating set of .Cn . A {0,1}-labelling .L of .Cn has the following properties. (i) .L can have at most one sequence of consecutive three 0’s, -0-0-0-. (ii) .L does not contain a sequence of -1-0-0-1-0-0-1-. (iii) If .L contains a sequence -1-0-0-0-1-, it can be followed by -1- or -0-1-, but not by -0-0-; also can be preceded by -1- or -1-0-, but not by -0-0-. (iv) One set of -1-0-0-0-1- combined with a series of repeated -1-0-1-0-0-’s (or -1-0-0-1-0-’s) would minimize the number of 1’s in a {0,1}-labelling. For a cycle .Cn with .n ≥ 10, let us construct a locating set with the smallest size based on the above properties. A {0,1}-labeling of .Cn with a locating set Fig. 1 .ln(G)=2, .ln(H )=4

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Table 1 {0,1}-Labellings of .Cn with minimum number of 1’s n = 5k n = 5k + 1 . n = 5k + 2 . n = 5k + 3 . n = 5k + 4

-1-0-1-0-0- . . . . -1-0-1-0-0- (-1-0-1-) -1-0-1-0-0- . . . . -1-0-1-0-0-0- (-1-0-1-) -1-0-1-0-0- . . . . -1-0-1-0-0-0-1- (-1-0-1-) -1-0-1-0-0- . . . . -1-0-1-0-0-0-1-0- -(1-0-1-) -1-0-1-0-0- . . . . -1-0-1-0-0-0-1-0-1- (-1-0-1-)

.

.

= 2k = 2k .ln(Cn ) = 2k + 1 .ln(Cn ) = 2k + 1 .ln(Cn ) = 2k + 2 .ln(Cn ) .ln(Cn )

must contain at least one sequence of -1-0-1-0-0-. Starting with one sequence of -1-0-1-0-0-, Table 1 displays possible {0,1}-labellings with minimum number of 1’s. The ‘ ... ’ represents a series of repeated -1-0-1-0-0-’s, and the ‘(-1-0-1-) ’ at the end of each {0,1}-labelling duplicates the labelling of the first three vertices in the first sequence of -1-0-1-0-0-. Although the above result is for cycles with at least 10 vertices, a similar reasoning can be applied to smaller cycles. Proposition 1 For a cycle .Cn , ⎧ ln(Cn ) =

.

2k

when n = 5k − 1, 5k, and 5k + 1

2k + 1 when n = 5k + 2, and 5k + 3,

where k is a positive integer. We can apply a similar argument for a path .Pn with n vertices to find its locating number. Although there are several cases of {0,1}-labellings to consider for the vertices at the beginning of a path, starting with a labelling of 0-0-1-0-1-0- combined with a series of repeated -1-0-1-0-0-’s leads us to the following result. Proposition 2 For a path .Pn , ⎧ ln(Pn ) =

.

2k

when n = 5k − 1, 5k, and 5k + 1

2k + 1 when n = 5k + 2, and 5k + 3,

where k is a positive integer. For the strictly locating number of cycles and path, one can easily verify sln(Cn ) = ln(Cn ) and .sln(Pn ) = ln(Pn ) for .n ≥ 7 since the degree of each vertex in a cycle or a path is at most 2.

.

3 Minimum Locating Number Suppose a simple connected graph .G = (V (G), E(G)) with n vertices has a locating S with size s. Let us consider a subgraph H of G which is portraying the ‘framework’ of the locating set S as follows:

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(i) .V (H ) = V (G), and (ii) .E(H ) = {(x, y) ∈ E(G) : x ∈ S, y ∈ V (G) \ S}. Then H is a bipartite subgraph of G with parts S and .V (G) \ S. This subgraph H may not be connected. From the definition of a locating set, .|V (H ) \ S| = |{N (u) ∩ S : u ∈ V (H ) \ S}|. Since .{N (u) ∩ S : u ∈ V (H ) \ S} is a subset of the power set of S, .|V (H ) \ S| ≤ 2s . Therefore .n = s + |V (H ) \ S| ≤ s + 2s . This leads us to the following lemma. Lemma 1 Let .G = (V (G), E(G)) be a connected graph with n vertices and S a locating set with s vertices. Then (i) G contains a bipartite subgraph H with parts S and .V (G) \ S, and .E(H ) = {(x, y) ∈ E(G) : x ∈ S, y ∈ V (G) \ S}. (ii) .n ≤ 2s + s Lemma 2 Let G be a graph with n vertices and s an integer satisfying .2s−1 + (s − 1) < n ≤ 2s + s. Then .ln(G) ≥ s. Proof We will prove by contradiction. Suppose G has a locating set T with size .t < s. By Lemma 3 (ii), .n ≤ 2t + t. From the hypothesis s satisfies .2s−1 + (s − 1) < n, and so .2s−1 + (s − 1) < 2t + t, which is a contradiction to .t < s. Suppose n and s are integers satisfying .2s−1 + (s − 1) < n ≤ 2s + s. Let S be a set with s elements. Since .2 ≤ s ≤ 2s−1 ≤ n − s ≤ 2s , we can consider a set T which is a subset of the power set of S and contains .n − s elements while satisfying that .{u} ∈ T for all .u ∈ S. Now let us define a graph M such that .V (M) = S ∪ T and .E(M) = {(x, y) : x ∈ y for some x ∈ S, y ∈ T } ∪ {(x, y) : x ∈ T , y ∈ T }. Then M is a connected graph with n vertices and S is a locating set of M. Thus we have the following theorem. Theorem 1 .min {ln(G) : G is a connected graph with n vertices} = s when n and s are integers satisfying .2s−1 + (s − 1) < n ≤ 2s + s. For the strictly locating number of graphs with n vertices, we can make a parallel argument with that of the locating number of graphs with n vertices. The only difference is that the bipartite subgraph H portraying the ‘framework’ of a strictly locating set S satisfies .|V (H ) \ S| ≤ 2s − 1, instead of .|V (H ) \ S| ≤ 2s . This yields the following corollary. Corollary 1 .min {sln(G) : G is a connected graph with n vertices} = s when n and s are integers satisfying .2s−1 + (s − 2) < n ≤ 2s + (s − 1). Moving Forward In Theorem 1, we established the minimum locating number of graphs with order n by observing the existence of a bipartite subgraph with a locating set as one of its parts. One would be interested in finding locating numbers similar to those of paths and cycles (as in Propositions 1 and 2) for somewhat ‘uniformly’ structured graphs, such as grids, hypercubes, or 2-dimensional trees. A set of {0,1}-labelled building blocks seems to exist for such graphs although finding

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them appears to be rather involved. We suspect that a modification of the method employed by Omega and Canoy [1] to obtain the locating number of the join of graphs could be combined with {0,1}-labelled building blocks for such graphs.

References 1. S.A. Omega, S.R. Canoy Jr., Locating sets in a graph, Applied Math. Sci. vol. 9 (2015) no 60, 2957–2964. 2. P.J. Slater, Leaves of trees, Congr. Numer., 14 (1975), 549–559. 3. D.F. Rall, P.J. Slater, On the locating domination number for certain classes of graphs, Congressus Numerantium, 45 (1984), 77–106. 4. S J. Seo, P.J. Slater, Open neighborhood locating-dominating sets, Australasian J of Comb, 46(2010), 109–119.

j -Multiple, k-Component Order Neighbor Connectivity Alexis Doucette and Charles Suffel

Abstract Consider a network modeled by a graph G on n nodes and e edges. There exist several parameters to determine the vulnerability of G. One such parameter is the domination number, which measures the minimum number of nodes necessary in a set so that its closed neighborhood is the entire graph. Multiple domination, sometimes referred to as j -domination, is a variety of domination that requires every node to either be in the set or adjacent to at least j nodes from it. The vulnerability parameter k-component order neighbor connectivity is an extension of domination and is defined as the minimum number of nodes that when removed along with their neighbors leave only components of order less than k. The new (k) parameter, j -multiple, k-component order neighbor connectivity, denoted .κnc,j , extends both concepts and is defined as the minimum number of nodes that need to be removed, along with their neighbors, such that the surviving subgraph contains only components of order at most k and every node outside of the set that is adjacent to it is adjacent to at least j nodes from it. The complexity of computing the value for this parameter is NP-hard since it coincides with j -domination when k is 1. Here we introduce this new parameter, establish its bounds, and compare it to several other (k) previously established parameters. We also establish formulas for .κnc,j for several classes of graphs, including complete and complete bipartite graphs as well as paths, cycles, wheels, and complete grid graphs. Keywords Domination · Multiple domination · Component order neighbor connectivity · Network vulnerability

A. Doucette (□ ) · C. Suffel Stevens Institute of Technology, Hoboken, NJ, USA © The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 S. Heuss et al. (eds.), Combinatorics, Graph Theory and Computing, Springer Proceedings in Mathematics & Statistics 462, https://doi.org/10.1007/978-3-031-62166-6_12

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1 Introduction Developed by Oystein Ore in 1962, the domination number, the minimum order of a set of nodes with the property that every node of the graph is either in or adjacent to the set, is a measure of network vulnerability [8]. Domination has been expanded in many ways over the years. Several varieties of domination have been developed including multiple domination, sometimes referred to as j -domination, which requires that nodes not in the j -dominating set are adjacent to at least j nodes from it for a given positive integer j [1, 4]. Another extension of domination, kcomponent order neighbor connectivity, was introduced by K. Luttrell et al. [6]. For a given threshold value k between 1 and n, this parameter measures the minimum number of nodes such that the removal of their closed neighborhoods leaves the graph in a failure state where every surviving component has order less than k. In the case when k is 1, this is equivalent to the domination number. The parameter j -multiple, k-component neighbor order connectivity combines the concepts of j -domination and k-component order neighbor connectivity. For this new parameter, a set D is considered a j -multiple, k-component order neighbor failure set, also referred to as a .j, k-failure set for short, provided that the removal of the nodes in D and their neighbors leaves only components containing less than k nodes and every node not in D that is adjacent to a node from D is adjacent to at least j nodes from D. The j -multiple, k-component neighbor order connectivity is the number of nodes in a smallest such set. When .j = 1, this parameter is identical to k-component order neighbor connectivity. Similarly, when .k = 1, the j -multiple, 1-component neighbor order connectivity is equivalent to j -domination. When both j and k are 1, this parameter coincides with domination. In this chapter, we establish this new parameter, compare it to its predecessors, and establish bounds. We also establish the parameter for several well-known classes of graphs.

1.1 Applications This new parameter has many potential applications. Just as k-component order neighbor connectivity has been used as a measure of network vulnerability, specifically of spy or terrorist networks, j -multiple, k-component order neighbor connectivity can also be used and may even give a better estimate of the minimum number of people necessary to capture to dismantle such a network. Under k-component order neighbor connectivity, a network is considered dismantled if everyone captured gave up all of their immediate contacts, leading to their capture, and the remaining parts of the organization were small enough to be deemed nonviable [6]. This does not take into consideration scenarios where someone captured does not give up all of their contacts. To account for this, it may be necessary to increase the level of domination, i.e., to have multiple sources connected to each of the contacts captured.

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This new parameter may also be useful in placement planning. For example, when planning the placement of subway stops in an urban environment, it may be desirable to have every corner in the city be close to multiple stops. Suppose it is sensible to have each corner located no more than 5 blocks away from 2 subway stops. The city layout can be modeled by a graph with nodes representing the corners and edges representing the blocks that connect them. Then, the problem of finding the minimum number of stops necessary to meet these parameters can be achieved by finding a minimum 2-multiple, 5-component failure set of the graph.

1.2 Terminology Herein, a graph .G = (V , E), or G, is a simple graph consisting of finite set V of n nodes and a possibly empty finite set E of two-element subsets of V . Since G is simple, G is undirected and contains no self loops or multiple edges. For standard graph-theoretic notation and terminology, we refer to the book “Graphs and Digraphs” authored by Chartrand et al. [2]. Definition 1 For a graph G, .S ⊆ V (G) is a dominating set of G if every node in V (G) is either in S or adjacent to a node in S. The domination number, denoted .γ (G), is the order of a smallest dominating set S of G. .

Definition 2 A set D is a multiple dominating set, sometimes referred to as a .j dominating set, if every node in .V (G)−D is dominated by at least j nodes from D for a fixed positive integer j . The order of a smallest such set is referred to as the .j domination number, denoted .γ j (G). The 2-domination number is often referred to as the double domination number. Definition 3 A node u is subverted or failed by a set D if u is adjacent to at least one element of D. Definition 4 For an integer .1 ≤ k ≤ n, .U ⊆ V is a failure set of G provided each component of .G−N [U ] has order at most .k −1. The order of a smallest such failure set is called the .k-component order neighbor connectivity of .G and is denoted by (k) (k) .κ nc (G) or .κ nc . In other words, the .k-component order neighbor connectivity of G is the minimum number of closed neighborhoods whose removal ensures that all remaining components have order less than some threshold value, k. Definition 5 Given integers .1 ≤ k ≤ n and .j ≥ 1, .U ⊆ V (G) is a .j -multiple, .kcomponent order neighbor failure set, sometimes referred to as a .j, k-failure set for short, of G provided that each component of .G − N[U ] has order at most .k − 1 and each subverted node in .V −U is subverted by at least j nodes in U . The order of a smallest such .j, k-failure set is referred to as the .j -multiple .k-component order (k) neighbor connectivity of .G and is denoted .κ (k) nc,j (G) or .κ nc,j .

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Definition 6 If every component of a graph has order less than k, we say that the graph is a failure state or in a failure state. Otherwise, we say that it is an operating state.

2 General Results Since the concept of a j -multiple, k-component order neighbor connectivity set extends both j -domination and k-component order neighbor connectivity, here we compare it to both parameters. It is easy to see that in the case of .j = 1, the parameter is equivalent to k-component order neighbor connectivity and when .k = 1, it is equivalent to j -domination. When both k and j are 1, it is equivalent to domination. This section demonstrates the relationship between these parameters, presents bounds for the parameter in terms of j and k, and provides necessary and sufficient conditions for the minimality of a .j, k-failure set. The example below demonstrates the values for each parameter for a particular graph and a minimum failure set for each such that .j = 2 and .k = 2 where applicable. A minimum failure set for each is depicted in black. Example 1 See Fig. 1.

(2) (2) Fig. 1 .γ (G), γ2 (G), κnc (G), and .κnc,2 (G)

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Observation 1 For a graph G on n nodes where .1 ≤ k ≤ n and .j ≥ 1, 1. .γj (G) ≥ min{j, n}. (k) 2. if .n ≥ j , then .κnc,j (G) ≥ j . (k)

3. if G is connected and .Δ(G) < j , then .κnc,j (G) = n. Proof 1. If there exists a node outside of the failure set, at least j nodes need to be in the failure set to dominate it. 2. Similarly, if there exists a node outside of the failure set, at least j nodes are necessary to dominate it. Since .n ≥ k, the failure set must be nonempty. 3. As above, the failure set is nonempty and since no node has j neighbors, no node can be subverted sufficiently. Therefore, every node must be in the failure set. We now demonstrate that j -multiple, k-component order neighbor connectivity is equivalent to j -domination when .k = 1 and equivalent to k-component order neighbor connectivity when .j = 1. (1)

Theorem 1 For all .j ≥ 1, .κnc,j (G) = γj (G). (k)

Proof Let .k = 1 and D be a failure set of G. By the definition of .κnc , D is a .j, 1-failure set of G provided that every component of .G − N [D] has order less than 1 and every subverted node in .V − D is subverted j times. Therefore, every component in .G − N [D] has order 0, meaning .G − N [D] = ∅ and every node not in D is subverted and, therefore, subverted j times. Therefore, D is also a j (1) dominating set, so .γj (G) ≤ κnc,j (G). Now let .D1 be a j -dominating set. Then, by the definition of a multiple dominating set, every node in .V − D is adjacent to at least j nodes from .D1 . It follows that .G − N [D1 ] = ∅ so every component has order .0 < k, ∀ .k ≥ 1. (1) (1) Therefore, .D1 is a .j, 1-failure set, so .κnc,j (G) ≤ γj (G). Hence, .κnc,j (G) = γj (G). (k)

(k)

Theorem 2 For .1 ≤ k ≤ n, .κnc,1 (G) = κnc (G). Proof Let D be a .1, k-failure set of G. Then, by definition, every component in (k) (k) .G − N [D] has order less than k. Thus, D is a failure set and .κnc (G) ≤ κ nc,1 (G). (k)

(k)

Now let .D1 be a .κnc failure set of G such that .κnc,j (G) = |D1 |. Then, by the (k)

definition of .κnc , every component in .G−N [D1 ] has order less than k. By definition of a subverted node, every subverted node is adjacent to at least 1 node in .D1 , (k) (k) (k) making .D1 a .1, k-failure set. Therefore, .κnc,1 (G) ≤ κnc (G). Hence, .κnc,1 (G) = (k) κnc (G). (k)

The inequalities below provide bounds for .κnc,j (G) with respect to different values of j and k, respectively. (k)

(k)

Theorem 3 .κnc,j1 (G) ≤ κnc,j2 (G) for all .1 ≤ j1 ≤ j2 and .1 ≤ k ≤ n.

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Proof Any .j2 , k-failure set is also a .j1 , k-failure set since the components of the subgraph induced by the removal of the closed neighborhood of the .j2 , k-failure set are all of order less than k and every subverted node is subverted at least .j2 times, which is more than the necessary .j1 times. Remark 2 In particular, the above result shows that the k-component order neighbor connectivity is always less than or equal to the j -multiple, k-component order (k) (k) neighbor connectivity. That is, .κnc (G) ≤ κnc,j (G) for all .j ≥ 1. (k )

(k )

1 2 Theorem 4 .κnc,j (G) ≤ κnc,j (G) for all .1 ≤ k2 ≤ k1 ≤ n and .j ≥ 1.

Proof Any .j, k2 -failure set is also a .j, k1 -failure set since all subverted nodes are subverted j times and all components of the subgraph induced by the removal of the closed neighborhood of the .j, k2 -failure set are all of order less than .k2 , so they are also of order less than .k1 . Remark 3 It follows directly from Theorem 4 that the j -multiple, k-component order neighbor connectivity of a graph is always less than or equal to its j (k) domination number. That is, .κnc,j (G) ≤ γj (G) for .1 ≤ k ≤ n. It was shown in [5] that if D is a minimum dominating set, then .V − D will contain at least node dominated no more than twice. We show that this result extends to higher values of k. Theorem 5 If .1 ≤ k ≤ n and D is a minimum .1, k-failure set of a graph G, then at least one node in .V − D is subverted by no more than two nodes in D. Proof Assume every subverted node in .V − D is subverted by at least 3 nodes in D. Let .u ∈ V − D be subverted by .{v, w} ∈ D. Then .D ' = D − {v, w} ∪ {u} is a ' .1, k-failure set with .|D | < |D|. (k) (k) Theorem 6 If .Δ(G) ≥ j ≥ 2 then .κnc,j (G) ≥ κnc (G) + j − 2 for all .1 ≤ k ≤ n.

Proof Let D be a minimum .j, k-failure set, .u ∈ V (G) − D and .{v1 , v2 , ...vj } be distinct nodes in D that subvert u. .V (G) − D /= φ since .Δ(G) ≥ j and therefore the set .V (G) take away one node of degree .Δ(G) is a .j, k-failure set of order less than .V (G). Since D is a minimum .j, k-failure set, each subverted .v ∈ V (G) − D is subverted by at least one element in .D−{v2 , ..., vj }. Since u is adjacent to each node in .{v2 , ...vj }, D ' = D − {v2 , ..., vj } ∪ {u} is a minimum failure set in G. Therefore (k) (k) (k) (k) ' .κnc (G) ≤ |D | = κ nc,j (G) − (j − 1) + 1. Hence .κnc,j (G) ≥ κnc (G) + j − 2. (k) (k) Corollary 1 If .Δ(G) ≥ j ≥ 3 then .κnc,j (G) > κnc (G) for all .1 ≤ k ≤ n.

The necessary and sufficient conditions for the minimality of a dominating set were previously established by Chartrand et al. in [2]. With the extra stipulations that every node in the failure set D must be either adjacent to less than j nodes from D and in a component of order at least k or there must be another node u in the neighborhood of v and not in D that has precisely j neighbors in the failure set for (k) .κ nc,j , this result carries over to higher values of j .

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Theorem 7 For all .j ≥ 2 and .1 ≤ k ≤ n, a .j, k-failure set D of a graph G is minimal if and only if one of the following holds for all .v ∈ D 1. .|N (v) ∩ D| < j and the component containing v in .V − (D − {v}) has order at least k, or 2. There exists .u ∈ V − D such that .|N (u) ∩ D| = j and .u ∈ N (v) Proof Suppose D is a .j, k-failure set of a graph G and one of the two conditions hold for all nodes in D. For a given v, if .|N (v) ∩ D| < j and the component containing v in .V − (D − {v}) has order at least k, then if you remove any node .u ∈ D, u will either not be adjacent to any nodes in D, in which case it is a part of a component of order at least k, meaning the graph is not in a failure state, or v is adjacent to less than j nodes in D. In either case, the node v is necessary for the failure of the graph and .D − {v} is not a .j, k-failure set. For a node .v ∈ D such that .∃ .u ∈ V − D such that .|N (u) ∩ D| = j and .u ∈ N (v), the set .D − {v} will not be a .j, k-failure set because the node u will not be subverted j times. Now suppose D is a minimal .j, k-failure set and .∃ .v ∈ D such that either .|N (u)∩ D| ≥ j or the component containing v in .V − (D − {v}) has order less than k and there does not exist .u ∈ V − D such that .|N (u) ∩ D| = j and .u ∈ N (v). Suppose we are in the first case. That is, there exists .v ∈ D such that .|N (u) ∩ D| ≥ j and there does not exist .u ∈ V − D such that .|N (u) ∩ D| = j and .u ∈ N (v). Consider the removal of v. Since D is a .j, k-failure set and for every node outside of the .j, k-failure set .|N (u) ∩ D| /= j , every node u in .N (v) − D is subverted more than j times. That is, j nodes in .D − {v} subvert every node in .N (v) − D. Since v is adjacent to j nodes from D, v is also subverted sufficiently by D. Therefore, .D − {v} is a .j, k-failure set, which contradicts the minimality of D. Next, suppose there exists .v ∈ D such that the component containing v in .V − D ∪ {v} has order less than k and there does not exist .u ∈ V − D such that .|N (u) ∩ D| = j and .u ∈ N(v). Then the node v is not require to be in the .j, k-failure set to create a component of order less than k and every .u ∈ V − D is either not adjacent to v or .|N (u) ∩ D − v| /= j . Therefore, since D is a .j, k-failure set, .|N (u) ∩ D − {v}| = 0 or .|N (u) ∩ D − {v}| ≥ j and .D − {v} is a .j, k-failure set, which contradicts the minimality of D.

3 Results for Specific Graph Types This section establishes the j -multiple, k-component order neighbor connectivity for several classes of graphs. To begin, we establish the j -multiple, k-component order neighbor connectivity for complete and complete bipartite graphs. From there, we establish the parameter values for paths, cycles, and wheels. Note that the results for .j = 1 for complete bipartite graph and path were previously established in [6] and are included here for completeness. We then establish results for an .m × n complete grid graph. Note that in the case of .m = 2, this is equivalent to the ladder graph .Ln on 2n nodes. Mohan et al. previously established double domination

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results for some complete grid graphs in [7], which were corrected by Shaheen et al. in [9]. Here we expand these results to higher values of both j and k.

3.1 Complete Graphs and Complete Bipartite Graphs Here we establish the j -multiple, k-component order neighbor connectivity of (k) complete and complete bipartite graphs. The results included below for .κnc , i.e., the case when .j = 1, were also featured in [3]. (k)

Theorem 8 For a complete graph .Kn where .n ≥ k ≥ 1 and .j ≥ 1, .κnc,j (Kn ) = min{j, n}. Proof If .n < j , then all nodes must be in the failure set D following Observation 1. If .j ≤ n, then any j nodes will constitute a failure set since every node is subverted j times and all surviving components will have order 0. Since the order of a nonempty k failure set must be at least j , .κnc,j = j. (k)

(k)

Now we first consider .κnc,j for .j = 1, i.e., .κnc . The next theorem was originally established by Luttrell in [6]. An alternative proof is given below. Theorem 9 Given a complete bipartite graph .Kp,q where .1 < p ≤ q and .1 ≤ k ≤ n, ⎧ (k) .κnc (Kp,q )

=

2 k=1

(1)

1 k≥2

Proof When .k = 1, a single node can only fail the other partite set. Therefore, at least two nodes are necessary. One node from each partite set constitutes a failure (1) set with respect to .κnc . When .k > 1, a single node fails all nodes in the other partite set and when removed along with its neighbors yields only isolates. (k)

Next we consider .κnc,j (Kp,q ) when .k = 1, i.e., .γj (Kp,q ). Theorem 10 Given a complete bipartite graph .Kp,q where .1 < p ≤ q and .j ≥ 1,

(1) .κ nc,j (Kp,q )

= γj (Kp,q ) =

⎧ ⎪ ⎪ ⎨min{p, 2j } q ⎪ ⎪ ⎩p + q

j ≤p pq

Proof When .j ≤ p, j nodes from the partite set of order p will dominate all of the nodes from the other partite set. In order to dominate the remaining nodes of the partite set of order j , they can either be added to the .j, 1-failure set or j nodes from the other partite set can be used. When .p < j ≤ q, since the degree of all nodes in the partite set of order q is less than j , they must all be in the .j, 1-failure set. When

j -Multiple, k-Component Order Neighbor Connectivity

163

(2)

Fig. 2 .κnc,5 (K4,7 ) = 5

j > q, every node in the graph has degree less than j and therefore must be in the j, 1-failure set.

. .

Finally, we consider results for the complete bipartite graph for .j ≥ 2 and .k ≥ 2. Theorem 11 Given a complete bipartite graph .Kp,q where .p ≤ q, 2 ≤ k ≤ n, and j ≥ 1,

.

⎧ (k) .κ nc,j (Kp,q )

=

j

j ≤q

p+q

j >q

(3)

Proof Since .k ≤ n = p + q, we know the minimum .j, k-failure set is nonempty and therefore must have at least j elements. If .j ≤ q, take j nodes from the set of q nodes to subvert all nodes in the set of p nodes. If .j > q, it is impossible to subvert a node j times so every node must be in the .j, k-failure set. Example 2 Consider the complete bipartite graph .K4,7 depicted below in Fig. 2. When .j = 5 and .k = 2, as shown in black, 5 nodes are necessary in the .5, 2-failure set.

3.2 Paths, Cycles, and Wheels (k)

This section focuses on establishing .κnc,j for paths, cycles, and wheels. Note that the result for the .j = 1 case of Theorem 12 was established by Luttrell in [6]. An alternate proof, which was featured in [3], is provided for this result. Theorem 12 Given a path .Pn of length n, where .n = (k + 2)l + r, 0 ≤ r < k + 2, and .1 ≤ k ≤ n, ⎧| | n ⎪ j = 1, r < k ⎪ k+2 ⎪ ⎪ ┌ ┐ ⎪ ⎪ ⎨ n j = 1, r = k, k + 1 (k) k+2 (4) .κ nc,j (Pn ) = ⎪┌ n+1 ┐ ⎪ j = 2 ⎪ 2 ⎪ ⎪ ⎪ ⎩n j ≥3

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A. Doucette and C. Suffel

vi

v1 k + 2 nodes

k + 2 nodes

r nodes

Fig. 3 A minimum failure set of .Pn

(2)

(2)

Fig. 4 .κnc (P13 ) = 3 while .κnc,2 (P13 ) = 7

Proof In the case of .j = 1, the path .Pn can be divided into l segments of length k + 2, and one of length r as shown in Fig. 3. Notice that the failure of .vi for .1 ≤ i ≤ l in each segment removes the last three nodes of the segment and no other node. This leaves a disjoint union of l paths of (k) length .k − 1, which is a failure state for .κnc,j . Thus, at least l nodes are needed to

.

(k)

fail the path .Pn . If .r < k, then the remainder block is in a failure state for .κnc,j and | | (k) n . If .r = k, or .k + 1 then one more node is needed to fail so .κnc (Pn ) = l = k+2 ┐ ┌ (k) n the remainder block and so .κnc (Pn ) = l + 1 = k+2 . In the case of .j = 2, for a node to be subverted twice, both of its neighbors would ┌ ┐need to be in the .2, k-failure set. If you have a set D of cardinality less than n+1 . , then either two consecutive nodes are not in D or one of the endpoints is 2 not in D and therefore┌ only ┐ subverted once. In either case, D is not a .2, k-failure . The set of alternating nodes, including the end nodes set. Therefore, .|D| ≥ n+2 2 ┌ ┐ ┌ ┐ ≤ |D| ≤ n+1 constitutes a failure set of this cardinality, giving . n+1 . 2 2 If .j ≥ 3, then every node needs to be in the .3, k-failure set since .j > Δ(Pn ). Example 3 This example demonstrates how an increase in j can affect the (2) parameter. Below, we see a minimum .κnc failure set and a minimum .2, 2-failure set depicted in black for the same path .P13 (Fig. 4). A cycle is created by adding an edge to a path that connects the first and last nodes. We now establish the parameter values for all .j ≥ 1 and .1 ≤ k ≤ n for this class of graph. Theorem 13 Given a cycle on n nodes and .1 ≤ k ≤ n,

j -Multiple, k-Component Order Neighbor Connectivity

⎧┌ ┐ n ⎪ ⎪ ⎪ ⎨┌ k+2 ┐ (k) n .κ (C ) = n nc,j 2 ⎪ ⎪ ⎪ ⎩n

165

j =1 j =2

(5)

j ≥3

Proof When .j = 1, we revert to the case of k-component order neighbor connectivity. Fix a starting point and label the nodes .u1 , ..., un clockwise. Break the cycle into groups of .k + 2 nodes. Every group requires at least one node in the failure set to create components of order less than k. The last grouping, which may contain less than .k + 2 nodes, will require at least one node, otherwise, there will be (k) a node with one of their two nearest neighbors in the failure set of .κnc more than distance k away, meaning it is in too large of a component. Since every node has degree 2, when .j = 2, every node is either in the minimum .2, k-failure set or adjacent to a node in the .2, k-failure set. Suppose there is a .2, kfailure set D that when its closed neighborhood is removed from the graph leaves a nonempty component. Then there exists a node .v ∈ N (D) that is adjacent to a node in that component so v is only subverted once and D is not a .2, k-failure set. Therefore, for any .2, k-failure set D, .N [D] = G. Since at least every other node ┌ ┐ (k) (Cn ) = γ2 (Cn ) = n2 . needs to be in the .2, k-failure set, .κnc,2 When .j ≥ 3, j > Δ(Cn ) = 2 so every node must be in the .3, k-failure set otherwise at least one subverted node is not subverted .j ≥ 3 times. Example 4 An example of a minimum .2, k-failure set for .1 ≤ k ≤ n for a cycle on 9 nodes can be seen in black in Fig. 5. It contains 5 nodes. Theorem 14 Given a wheel .Wn on n nodes and .1 ≤ k ≤ n, ⎧ ⎪ 1 ┌ ⎪ ⎪ ┐ j = 1 ⎪ ⎪ n−1 ⎪ ⎪ 1+ 3 j = 2, n /= 5 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨2 ┌ ┐ j = 2, n = 5 (k) n−1 .κ j = 3, n /= 3 nc,j (Wn ) = ⎪1 + 2 ⎪ ⎪ ⎪ ⎪3 j = 3, n = 3 ⎪ ⎪ ⎪ ⎪ ⎪ n−1 4≤j ≤n ⎪ ⎪ ⎪ ⎩n j ≥n

(k) Fig. 5 .κnc,2 (C9 ) = 5

(6)

166

A. Doucette and C. Suffel

(k)

Fig. 6 .κnc,2 (W10 ) = 4

Proof In the case of .j = 1, the center node subverts every other node. When .j = 2, a .2, k-failure set D can either include the center node or not. If the center node is included, then it subverts every node so every node has to either be in the .2, k-failure set or adjacent to another member of the .2, k-failure set, i.e., 1 out of every 3 consecutive nodes needs to be in the .2, k-failure set otherwise┌ you ┐end up with a subverted node that is subverted less than j times so .|D| = 1 + n−1 . If 3 D does not contain the center node, then 1 out of every 2 consecutive nodes needs to be in the .2, k-failure ┐set otherwise ┐ at least ┌ one ┐ node will only be subverted once. ┌ ┌ n−1 n−1 n−1 Therefore, .|D| = 2 . .1 + 3 ≤ 2 except for when .n = 5. In the case of .j = 3, a subverted node needs to be subverted 3 times, twice by other nodes in the┌ cycle ┐ and once by the central hub. Following the argument above, nodes when .n /= 3. When .n = 3, all 3 nodes are necessary. this requires .1 + n−1 2 When .4 ≤ j < n, the only node of degree j is the central hub so every other node is necessary in the .j, k-failure set. The set of all nodes from the cycle constitutes a .j, k-failure set of order .n − 1. (k) When .j ≥ n, every node needs to be in the .j, k-failure set so .κnc,j (Wn ) = n. Example 5 We now consider the case of when .j = 2 and .1 ≤ k ≤ n. We see that for a wheel on 10 nodes, .W10 , a minimum .2, k-failure set will require 4 nodes as shown in black in Fig. 6.

3.3 Complete Grid Graphs Complete grid graphs are some of the most useful models for urban planning and studying the vulnerability of grid networks such as electric grids. An .m×n complete grid graph is defined as the Cartesian product of two paths, .Pm × Pn where .m ≤ n. Results for the double domination number of several complete grid graphs were first established by Mohan et al. in [7] and corrected by Shaheen et al. in [9]. In this section, we extend their result for double domination of .P3 × Pn to the 2-

j -Multiple, k-Component Order Neighbor Connectivity

167

multiple, k-component order connectivity of the graph. We also establish values for (k) κnc,j (Pm × Pn ) for all .3 ≤ m ≤ n for all .j ≥ 3.

.

Lemma 1 For a complete grid graph .G = Pm × Pn , where .k > (m − 1)2 , .3 ≤ m ≤ n, and D is a .2, k-failure set of G, the maximum order of a component in 2 .V (G) − N [D] is .(m − 1) . Proof Let D be a .2, k-failure set of G. If a component of .G − N [D] is at one of the ends of the grid graph, we claim that it can have order at most . (m−2)(m−1) . 2 Label the nodes .ux,y where x represents the node’s row and y represents the node’s column. Observe that at least one node in the first and last columns needs to be in D. To maximize the number of nodes in a component of .G − N [D], without loss of generality, consider the case of the upper left corner, .u1,1 being in D. That means, at most .m − 2 nodes, .{u3,1 , ..., um,1 }, from column 1 can be in the leftover component. .u3,2 must be in D and the nodes from each column in the leftover component must be consecutive otherwise there will be subverted nodes between them, which is not possible since only .u1,2 from column 1 is subverted. Therefore, only .m − 3 nodes from column 2 can be in the component. Following this logic, the next column can only have .m − 4 nodes and so on until we reach column .m − 2, which can only have one node in this component. The order of this component is m−2 Σ .

x=1

x=

(m − 2)(m − 1) < (m − 1)2 . 2

For a component of .G − N [D] that does not contain nodes from the first or last column, we claim that it can have order at most .(m − 1)2 . Suppose a component of .G − N [D], .G' , contains nodes from both the first and last row. Without loss of generality, consider the scenario where .G' contains x nodes, .{u1,y , ..., u1,y+x−1 }, from the first row. Then, .G' cannot contain .u2,y and .u2,y+x−1 since they are neighbors of .u1,y−1 and .u1,y+x and it would preclude them from being subverted twice. Furthermore, .G' needs to contain all of the nodes .u2,y+1 , ..., u2,y+x−2 since they cannot be subverted sufficiently since each only has one neighbor that is not in the neighborhood of .G' . Assume this holds for row p. That is, if a component contains x nodes from row p, it can contain at most .x − 2 nodes, everything but the end points of the component in the previous row, from row .p + 1. We aim to show that the same holds for row .p + 2. Suppose nodes .{up+1,w , ..., up+1,w+x−3 } are in .G' . From the previous step, we know .{up,w−1 , ..., up,w+x−2 } are also in ' .G . Therefore, nodes .up+1,w−1 and .up+1,w+x−2 are subverted twice, which means .up+2,w−1 and .up+2,w+x−2 need to be in D so .up+2,w and .up+2,w+x−3 are subverted and therefore not in .G' . Lastly, we claim that if D is a .2, k-failure set, then a component .G' of .G − N [D] that contains nodes from the first row cannot contain any nodes from the last row, row m. Suppose .G' does contain nodes from both the first and last row. Let .um,i be the leftmost node in the component in row m. Then, as previously established, ' .um−1,i−1 is also in .G . Thus, node .um,i−1 , the node to the left of .um,i , has degree

168

A. Doucette and C. Suffel

3. Since two of its neighbors are .um,i and .um−1,i−1 , both of which are not in D, it cannot be subverted sufficiently, which yields a contradiction. Combining this with the fact that if row i of .G' contains x nodes, then .G' must contain .x + 2 nodes from row .i − 1 and .x − 2 nodes from row .i + 1, and the last row of .G' must contain precisely 1 node. Therefore, .G' can contain at most 1 node from row .m − 1, 3 nodes from row .m − 2, and so on until reaching row 1 which will contain .2m − 3 nodes. Therefore, .G' can contain at most .1 + 3 + 5 + .. + 2m − 3 = (m − 1)2 nodes. Using the above lemma, we now establish results for .P3 ×Pn for any .n ≥ 3 when j = 2.

.

Theorem 15 For a complete grid graph .P3 ×Pn consisting of 3 rows and n columns where .n ≥ 3 and .1 ≤ k ≤ n,

(k)

κnc,2 (P3 × Pn )

.

⎧┌ ┐ n ⎪ ⎪ n+ 3 ⎪ ⎪ ⎪ ⎪ n ⎪ ⎪ ⎪ ⎪ ⎨n + 1

k=1 k = 2, n odd k = 2, n even

n | k = 3, 4 ⎪ ⎪ | ⎪ ⎪ n−3 ⎪ ⎪n − 4 k ≥ 5, n /≡ 0 mod 4 ⎪ ⎪ | | ⎪ ⎪ ⎩n + 2 − n−3 k ≥ 5, n ≡ 0 mod 4 4

(7)

Proof The case of double domination was proved by Mohan in [7] and corrected by Ramy Shaheen, Suhail Mahfud, and Khames Almanea in “On the 2-Domination Number of Complete Grid Graphs” [9]. Let D be a .2, k-failure set and .2 ≤ k ≤ n. We claim that the order of a component in .V − N[D] is 1, 2, or 4. Suppose a component in .V − N[D] contains three nodes, u, v, and w. Then, every neighbor of .u, v, and w that is outside of the component must be subverted by at least 2 nodes from D. If .u, v, and w are in the same row such that v is between u and w, then a node neighboring v cannot be subverted sufficiently since two of its neighbors are also subverted and it has degree 3 or 4. Without loss of generality, suppose u is on the top row of the grid graph and w is directly below it. Then, if v is adjacent to u, the node on the other side of u must be subverted twice, but it only has degree at most three and one of its neighbors is adjacent to w and therefore must be outside of D. If v is adjacent to w, then the node above v needs to be subverted twice, but it only has degree at most 3, including u and v, making that impossible. Therefore, a component in .V −N [D] cannot contain exactly 3 nodes. We also claim that a surviving component of .V −N [D] of order 2 cannot contain nodes from the middle row in any column. If the component consists of u from the top row and w from the middle row, then both of u’s neighbors can only be adjacent to at most one node from D since w’s neighbors both need to be subverted. If u and w are both in the middle row, then their neighbors have degree 3. Again

j -Multiple, k-Component Order Neighbor Connectivity

169

u' s neighbors cannot be subverted twice since both u and w’s neighbors cannot be in D. Furthermore, if two nodes on the top row are in .V − N[D], the nodes on the bottom row must be in the failure set in order to sufficiently subvert their neighbors. Let .k = 3 or 4 and suppose D is a .2, k-failure set such that .|D| < n. Then .∃ .m > 1 columns that do not contain any nodes from D and .n < m columns with 2 or 3 nodes in D. Label the nodes of .P3 × Pn as .ui,𝓁 where i denotes the row of the node and .𝓁 denotes the column. Suppose column .c𝓁 does not share any nodes with D. It is impossible for a component in .V − N [D] to contain more than one node and not have any nodes of D in its column so it is only necessary to consider the case where the order of a component in .v − N[D] is one. If this is the case for .c𝓁 , at least two of the nodes in .c𝓁 must be subverted, which means both of both nodes’ neighbors need to be in D. That is .c𝓁−1 and .c𝓁+1 both contain at least 2 nodes from D. Therefore, it is not possible that .n < m, a contradiction. This means that there is (k) at least one node in D for every column of the grid. Thus, .κnc,2 (P3 × Pn ) ≥ n for .k = 3, 4. Let n| be odd | and .k ≥ 2. Consider the set .D = {u1,𝓁 |𝓁 = 1 + 4x, x = .

0, 1, ...,

|

n−1 4

|

} .∪ .{u2,𝓁 |𝓁 = 2x + 1, x = 0, 1, ..., n−1 2 } .∪ .{u3,𝓁 |𝓁 = 3 + 4x, x =

}, which contains n nodes. .u1,𝓁 and .u3,𝓁 , .𝓁 even, are all subverted 0, 1, ..., n−2 4 twice, as is every node .u2,𝓁 , 𝓁 odd. Every remaining node is either in D or not adjacent to any nodes from D and only adjacent to subverted nodes, making D a (k) .2, k-failure set. Thus, .κ nc,2 = n for .k = 2, 3, 4 and odd n. Let n be even and .k > 2. Consider .D1 as D is described above for .P3 × Pn−1 except for the last 3 columns. Shift nodes in the failure set in the last 3 columns over to the right one and then set column .n − 2, the last unchanged column, to be the same as column .n − 3, as shown in Figs. 7 and 8 above. This again gives a failure (k) set on n nodes, so .κnc,2 = n for .k = 3, 4 and even n. Now, if n is even and .k = 2, the maximum order of a component allowed in .V − N[D] is one so the above set is not a failure set. Suppose there exists a failure

Fig. 7 A minimum failure set for .P3 × P13

Fig. 8 A minimum failure set for .P3 × P14

170

A. Doucette and C. Suffel

set F on n nodes. As shown above, the only way for a column to not contain any nodes of F is if both of its adjacent columns contain 2 nodes of F . A .2, 2-failure set constructed this way will never contain less than n nodes. In fact, if n is even, it will contain at least .n + 1 nodes since the end columns cannot have an empty intersection with F since they contain nodes of degree 2. Therefore, since there are an even number of columns, more than n nodes will be necessary. The other option for a .2, 2-failure set, as previously described, requires at least one node from each column. For this set to have degree n, it would need to consist of exactly one node per column. Since the corner nodes have degree 2, this is only possible if one corner in the column is in F and the other is in .V − N[F ] otherwise both of their neighbors in the adjacent column would be necessary in F . Since .k = 2, any failure set that contains one node in each column will require every other node in the center row to subvert the nodes on the top and bottom rows. Since n is even, one of the end columns will have its center node in F , which in turn would require two of the neighboring column’s nodes to be in F in order to subvert the (2) corners twice. This presents a contradiction. Therefore, .κnc,2 (P3 ×Pn ) = n+1 when n is even. When .k ≥ 5, by Lemma 1, the maximum order of a component in .G − N [D] is 4. Every column of these triangular components except for the ones that have 2 nodes in V-N[D] will need to contain at least one node from the failure set D, otherwise, there will be a node in the row above or below that needs to be subverted but only has 1 neighbor in D. Furthermore, we claim that a column without any nodes in D can only occur once per each grouping of 4 columns. Without loss of generality, assume column i shares no nodes with D and the bottom node .u3,i is subverted for .4 ≤ i ≤ m − 3. Then in order for the component in the subgraph induced by the removal of D to have order 4, as described above, .u2,i must be in it. This means both of .u3,i ’s neighbors, as well as the neighbors of .u2,i , are subverted. Furthermore, .u1,i−1 , u1,i , and .u1,i+1 must be in the surviving component so each of .u1,i−1 and .u1,i+1 ’s neighbors must also be subverted. These nodes only have degree three so both of their other neighbors, .u1,i−3 , u1,i+3 , u2,i−2 , and .u2,i+2 , also need to be in D. Therefore, each of the three columns on either side | of|column i must (k) contain at least one node from D. Therefore, .κnc,2 (G) ≥ n − n−3 . 4 (k)

When .n ≡ 3 mod 4, this provides the exact value for .κnc,2 . Consider the set |n| , u2,6+8i , u3,7+8i |i = .D = {u1,1+8i , u2,2+8i , u3,3+8i |i = 0, 1, ..., 4 } .∪ .{u1,5+8i| | |n| (k) 3 so .κnc,2 (G) = 0, 1, ..., 8 − 1}. D is a failure set of order . 4 (n − 1) = n − n−3 4 | | n − n−3 . 4 When .n /≡ 3 mod 4, there are . n−1 4 groupings of 4 nodes, but one of them will not be able to contain a column with no nodes in D because the last grouping will not have the necessary three columns to the right to subvert it. This means that at most n−1 no nodes with D and thus, the failure set must contain . 4 − 1 columns can share | | n−1 nodes. at least .n − 4 + 1 = n − n−3 4

j -Multiple, k-Component Order Neighbor Connectivity

171

For .n ≡ 1 mod 4, we consider the cases of .n ≡ 1 mod 8 and .n ≡ 5 mod 8 separately. | | If .n ≡ 1 mod 8, consider a set .D = {u1,1+8i | |, u2,2+8i , u3,3+8i |i = 0, 1, ..., n4 −2} .∪ .{u1,5+8i , u2,6+8i , u3,7+8i |i = 0, 1, ..., n8 −1} .∪ .{u2,n−1 , u3,n } | | . Similarly, if .n ≡ 5 mod 8, consider a set .D = is a .2, k-failure set of order .n− n−3 4 | | {u1,1+8i , u2,2+8i , u3,3+8i |i = 0, 1, ..., n4 − 1} .∪ .{u1,5+8i , u2,6+8i , u3,7+8i |i = | | | | 0, 1, ..., n8 − 1} .∪ .{u2,n−1 , u1,n }, which is a .2, k-failure set of order .n − n−3 . |4n | If .n ≡ 2 mod 8, consider a set .D = {u1,1+8i , u2,2+8i , u3,3+8i |i = 0, 1, ..., 4 − | | 2} .∪ .{u1,5+8i , u2,6+8i , u3,7+8i |i = 0, 1, ..., n8 −1} .∪ .{u1,n−2 , u2,n−1 , u3,n }, which | | is a .2, k-failure set of order .n− n−3 . Similarly, if .n ≡ 6 mod 8, consider a set .D = 4 | | {u1,1+8i , u2,2+8i , u3,3+8i |i = 0, 1, ..., n4 − 1} .∪ .{u1,5+8i , u2,6+8i , u3,7+8i |i = | | | | 0, 1, ..., n8 − 1} .∪ .{u1,n−2 , u2,n−1 , u1,n } is a failure set of order .n − n−3 . | 4 | (k) Therefore, for .k ≥ 5, and .n /≡ 0 mod 4, we get .κnc,2 (P3 × Pn ) = n − n−3 . 4 Finally, when .n ≡ 0 mod 4, we claim that at least one additional node is necessary in D than was required for the case when .n /≡ 0 mod 4. As previously stated, at most . n−1 4 − 1 columns can share no nodes with D, meaning .|D| ≥ n−1 n − 4 + 1. Here, we claim that for the last grouping of 4 columns, either 2 nodes in one column are necessary or one less column earlier in the graph can contain no nodes from D. Suppose there are . n4 − 1 columns that do not contain any nodes from D. This can only happen if a .2, k-failure set is constructed as described above, which means that D has a node in the column adjacent to the last grouping in either the first or last row. In order to subvert its adjacent node in the last column, the center node is required to be in D, which would also subvert the other corner, requiring a second node from the previous column to also be in D. Similarly, if instead you choose to add the adjacent node to the .2, k-failure set, that would subvert the center node of the last column, which is not adjacent to any nodes of D and will therefore require at least one more node in the .2, k-failure set. The only way for the last two columns to each contain one node is if the last column follows the pattern described above and ends in a corner. This would require shifting the pattern over two, which would mean 2 groupings would not be able to (k) have columns with no nodes in D. Therefore, when .n ≡ 0 mod 4, .κnc,2 (G) ≥ | | | | + 2. Consider the set .D = {u1,1+8i , u2,2+8i , u3,3+8i |i = 0, 1, ..., n4 } .∪ n − n−3 4 |n| .{u1,5+8i , u2,6+8i , u3,7+8i |i = 0, 1, ..., 8 − 1} .∪ .{u1,n−1 , u3,n−1 , u2,n }. Note that one of .{u|1,n−1|, u3,n−1 } is included in D. This gives a .2, k-failure set of G of order n+2−

.

n−3 4

for .k ≥ 5 and .n ≡ 0 mod 4.

We now establish the j -multiple, k-component order neighbor connectivity of a complete grid graph when .j = 3. Theorem 16 For an .m×n grid graph .Pm ×Pn where .2 < m ≤ n and .2 ≤ k ≤ mn,

172

A. Doucette and C. Suffel

┐ | | ⎧ ┌ | | m−2 ⎪ 2 n+1 + f (n) + 2 n2 + 2 ⎪ 2 4 ⎪ ⎪ ┐ | | ┌ ⎪ | | ┌ ┐ ⎪ m−2 ⎪ ⎨2 n+1 + f (n) + 2 n2 + n4 + 1 2 4 (k) ┐ | | | | | | ┌ .κ nc,3 (Pm × Pn ) = ⎪ ⎪ + m−2 f (n) − n+2 + n2 + 1 2 n+1 ⎪ 2 4 4 ⎪ ⎪ ┐ | | ┌ ⎪ ⎪ ⎩2 n+1 + m−2 f (n) + | n | 2 4 2

m ≡ 0 mod 4 m ≡ 1 mod 4 m ≡ 2 mod 4 m ≡ 3 mod 4 (8)

where ⎧ | | ┌ ┐ ┌ ┐ ⎨2 n + n + n+2 + 2 n ≡ 1 mod 4 2 4 4 ┐ .f (n) = | | ┌ ┐ ┌ ⎩2 n + n + n+2 + 1 n ≡ / 1 mod 4 2 4 4

(9)

Proof Given a failure set D, the maximum order of a component in .V − N [D] is one. Otherwise, there would be two adjacent nodes that are both subverted, each being adjacent to a node of .V − N [D], meaning they only have two neighbors that could be in the failure set, and not the necessary three. At least every other node of the first and last rows are necessary in D since they all have degree 3, except for the corner ┌ ┐ nodes, which have degree 2 and, therefore, must be in D. This is equivalent to . n2 nodes from each of these rows being required in D. We also note that the nodes in columns 1 and n meet these conditions and therefore must be treated the same. We claim that the order of any component in the surviving subgraph induced by the removal of the failure set D is one. Consider a component of order greater than one in .< V − N [D] >, then either there are two neighboring nodes that are both adjacent to subverted nodes or there are two nodes adjacent to subverted nodes that share a subverted neighbor. In either case that subverted node, which has degree 3 or 4, is adjacent to at most two nodes of D since two of its neighbors are either in .< V − N [D] > or are subverted themselves. Therefore, the largest component in .V − N [D] is an isolate. Additionally, we claim that any node in .V − N [D]’s nodes nearest neighbor in .V − N [D] is at minimum distance 4 away in G. Suppose node .ux,y is in .V − N [D] where x represents the node’s row and y its column. Then, .ux,y ’s neighbors, .ux−1,y , ux+1,y , ux,y−1 , and .ux,y+1 , all need to be subverted, which means that the rest of their neighbors need to be in the failure set. That is, .ux−2,y , ux−1,y−1 , and .ux,y+1 are necessary to subvert .ux−1,y , while .ux+2,y , ux+1,y−1 , and .ux+1,y+1 are necessary to subvert .ux+1,y , while .ux−1,y−1 , ux+1,y−1 , and .ux,y−2 are necessary to subvert .ux,y+1 , and .ux−1,y+1 , ux+1,y−1 , ux,y+2 are necessary to subvert .ux,y+1 . Since any node adjacent to a node of the .3, k-failure set is subverted and the above set comprises all nodes distance 2 from .ux,y , the minimum distance between nodes isolated in .V − N [D] is 4. Furthermore, each grouping of 4 nodes will need to contain at least one node from D. This indicates that for every grouping of 4 nodes in a row, at least one of the nodes is in the .3, k-failure set.

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Furthermore, as previously described, if a node is not in .N [D], then every other node from the row below and the row above it is required to be in the .3, k-failure set since the nodes directly above and below need to be three times and only | | subverted have 3 other neighbors. This gives us that at least . n2 nodes are necessary in D in every other row. Since the first row that can have a node outside of .N [D] is row 3, these will be the even numbered rows. Furthermore, since any other node in .V − N [D] must be at least distance 4 away, the nearest node in .V − N [D] two rows below must also be either 2 to the right or 2 to the left. Note that this row may also be such that one out of every 4 nodes is in D. Thus, for each complete grouping of 4 rows, starting with row 2, at least | n | ┌ n ┐ ┌ n+2 ┐ .2 are necessary in D. + + 2 4 4 We consider the different congruence classes of m and n modulo 4 separately. When n is even, since the last two columns must contain every other node, the rows that contain nodes outside of the closed neighborhood of the failure set will not have any nodes outside of .N [D] in the last two columns. That is, when n is even, the ┌ n ┐rows that have a node in column 3 that is outside of the failure set will require rows that have a node outside of . 4 + 1 nodes. Similarly, when .n ≡ 2 mod┌ 4, the ┐ n+2 the failure set in column 5 will also require . 4 + 1 nodes. When m is odd, if the pattern described above is used, the second to last row will have every other node in the failure set. When .m ≡ 1 mod 4, the last grouping of rows, excluding only one of which will need ┌ ┐ the last row, will contain three ┌ rows, ┐ to contain . n4 nodes when .n /≡ 0 mod 4 and . n4 + 1 nodes when it is. Therefore, ┐ ┌ when .n ≡ 1 mod 4, in total we have at least .2 n+1 nodes from rows 1 and m, 2 | | | ┌ ┐ ┌ ┐ | n n n+2 m−2 [2 + 4 + 4 + 2] from each of the groupings of 4 rows, and . | 4n | ┌ n 2 ┐ .2 .n /≡ 1 mod 4, in 2 + 4 + 1 nodes┌ from ┐the last grouping to be in D.|When | | | ┌ ┐ n+1 m−2 total we have at least .2 2 nodes from rows 1 and m, . 4 [2 n2 + n4 + ┐ ┌ | | ┌ ┐ n+2 + 1] from each of the groupings of 4 rows, and .2 n2 + n4 + 1 nodes from 4 the last grouping need to be in D. When .m ≡ 3 mod 4, the last grouping of rows, |excluding the last row, will | contain only one row, which will require no less than . n2 nodes to fail. Therefore, ┌ ┐ when .n /≡ 0 mod 4 and . n4 + 1 nodes when it is. Therefore, when .n ≡ 1 mod 4, ┐ | | | | ┌ ┐ ┌ m−2 [2 n2 + n4 + in total we have at least .2 n+1 nodes from rows 1 and m, . 2 4 ┐ ┌ | | n+2 + 2] from each of the groupings of 4 rows, and .2 n2 nodes from the last 4 ┐ ┌ grouping to be in D. When .n /≡ 1 mod 4, in total we have at least .2 n+1 nodes 2 | | | | ┌ ┐ ┌ ┐ [2 n2 + n4 + n+2 + 1] from each of the groupings from rows 1 and m, . m−2 4 |n| 4 of 4 rows, and .2 2 nodes from the last grouping to be in D. When m is even, the second to last row is an odd-numbered row, meaning the row before it has every other node in D, and needs to contain every other node in order to sufficiently subvert the nodes of row m not in the failure set. As such, if

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| | groupings of 4 rows, each containing m ≡ 2 mod 4, then the graph contains . m−2 4 | n | ┌ n ┐ ┌ n+2 ┐ | n | ┌ n ┐ ┌ n+2 ┐ +2 +1 nodes nodes if .n ≡ 1 mod 4 and .2 .2 + + 2 4 4 2 + 4 + 4 if .n /≡ 1 mod 4. Since the last row of the last grouping, which normally contains isolated nodes in the other groupings cannot | have | that here, we must remove this row from our calculation and replace it with . n2 . Therefore, a failure set of the graph ┐ ┌ ┐ | | ┌ | n | ┌ n ┐ ┌ n+2 ┐ m−2 n+2 + + 2] − + n2 − 1 [2 + + must contain at least .2 n+1 2 4 2 4 4 4

.

nodes when .n ≡ 1 mod 4. As described above, if .n /≡ 1 mod 4, an additional . m−2 4 nodes will be required in D. | | complete groupings of 4 If .m ≡ 0 mod 4, then the graph contains . m−2 4 rows and the last grouping contains three rows excluding the second to last row. Again since m is even, the third to last row and second to last row both need to contain Therefore, a failure set of the graph must contain at least ┐every other|node. ┌ | | | ┌ n ┐ ┌ n+2 ┐ m−4 n n+1 + + 2] + 2 n2 + 2 nodes when .n ≡ 1 mod + [2 .2 2 4 2 4 + 4 4. As described above, if .n /≡ 1 mod 4, an additional . m−4 4 nodes will be required in D. | | ,1 ≤ When .m ≡ 3 mod 4,consider the set .D = {u1+4i,k |i = 0, ..., n−2 4 |n| |m| k ≤ n, k odd} .∪ .{u2i,2k |i = 1, ..., 2 , k = 1, ..., 2 } .∪ .{u3+4i,1+4k |i = | | | | ┌ ┐ ┌ ┐ , k = 1, .., n4 } .∪ .{u5+4i,3+4k |i = 0, ..., m−2 , k = 1, .., n4 }. 0, ..., m−2 4 4 | | } to D. If .n ≡ 3 mod 4, add If .n ≡ 1 mod 4, also add .{u3+4i,n |i = 0, .., m−2 4 | | | | m−2 m−2 } } to D. Then, for all .{u3+4i,n−2 |i = 0, .., and .{u5+4i,n |i = 0, .., 4 4 values of n, D is a failure set of the same order mentioned above, giving us equality. When .m ≡ 1 mod 4, consider a set D as described above except for rows .m − 1 and m. Add in every even-numbered node from row .m − 1 and every odd-numbered node from row m, as well as the last node in row m if it is not already included. This will give a failure set of the order mentioned above for all n, giving us equality. When .m ≡ 0 mod 4, consider the set D as described above except for rows m and .m − 1. Take this set together with every odd-numbered node from row .m − 1 and every even-numbered node from row m, as well as the first and last node in row m if it is not already included. This will provide a failure set of the order mentioned above, giving us equality. When .m ≡ 2 mod 4, again consider the set D as described above except for rows .m − 2 and m. Take this set together with nodes all odd-numbered nodes from row .m − 2 and all even-numbered nodes from row m, as well as the first and last node in row m if it is not already included. This will provide a failure set of the order mentioned above, again giving us equality. Example 6 Figure 9 provides an example of a minimum failure set for a .9 × 13 (k) grid graph. Here, we can see that .κnc,3 = 68 for .2 ≤ k ≤ n and the nodes for such a minimum .3, k-failure set are depicted in black.

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Fig. 9 Minimum failure set of .P9 × P17

We now establish results for a complete grid graph when .j = 4. Note that the parameter does not depend on k and, therefore, the equation below also gives the 4-domination number of an .m × n complete grid graph. Theorem 17 For an .m × n grid graph .Pm × Pn with .2 ≤ m ≤ n and .1 ≤ k ≤ n, | (k)

κnc,4 (Pm × Pn ) = 2(m + n − 2) +

.

| (m − 2)(n − 2) . 2

(10)

Proof Let D be a .4, k-failure set of .G = Pm × Pn . Then .V (G) = N [D] since the graph is connected and every node has degree at most 4, which results in no subverted node being capable of being adjacent to a node outside of D. Furthermore, no two adjacent nodes can be in .V (G)−D. There are .2m+2n−4 nodes on the outer edges of the grid, all of which have degree at most 3, and consequently need to be in the failure set since they cannot be subverted 4 times. This leaves | .(m−2)(n−2) | inner nodes, no two of which can be adjacent, which means at least . (m−2)(n−2) of these 2 | | inner nodes must in the failure set. Therefore, .|D| ≥ 2m + 2n − 4 + (m−2)(n−2) . 2 A failure set of this order can be constructed as follows. Let i represent the row and j represent the column of a node n. Let .S = U .vij , fpr .i = 1, ..., m and .j = 1, ...,U {vij |i = 1, m and j = 1, ..., n} {vij |i = 2, ..., m − 1Uand j = 1, n} {vij |2 ≤ i ≤ m − 1 for .i even and 3 ≤ j ≤ n − 1 for .j odd} {vij |3 ≤ i ≤ m − 1 for .i odd and 2 ≤ j ≤ n − 1 for .j even}. Since .N [S] = G and .|N (u) ∩ S| = 4 for all | | (m−2)(n−2) .u ∈ V (G)−S, S is a .4, k-failure set and the order of S is .2m+2n−4+ . 2 | | Therefore, .|D| ≥ 2m + 2n − 4 + (m−2)(n−2) , giving us equality. 2 The next observation follows since .Δ(Pm × Pn ) = 4.

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Fig. 10 A minimum .4, k-failure set of .P7 × P13 contains 63 nodes for .1 ≤ k ≤ n

(k)

Observation 2 When .j ≥ 5 and .1 ≤ k ≤ n, .κnc,j (Pm × Pn ) = mn. Example 7 Given a .7 × 13 complete grid graph and .j = 4, for any .1 ≤ k ≤ n, this graph will requires 63 nodes to fail. A minimum .4, k-failure set, which is also a minimum 4-dominating set, is shown in black (Fig. 10).

References 1. Alavi, Y., Chartrand, G., Lesniak, L., Lick, D.R., and Wall, C.E. Graph Theory with Applications to Algorithms and Computer Science. American Mathematical Society Colloquium Publications, 1962. 2. Chartrand, G., L. Lesniak, and P. Zhang. Graphs and Digraphs. Chapman and Hall/CRC, 5th edition, 2011. 3. Doucette, A., A. Muth, and C. Suffel (2018). An exceptional invariant, k-component order neighbor connectivity, A Generalization of Domination Alteration Sets in Graphs. Congressus Numerantium 230, 167–190. 4. Fink and Jacobson. n-domination in graphs. Graph Theory with Applications to Algorithms and Computer Science, pages 283–300, 1985. 5. Haynes, T. W., S. T. Hedetniemi, and P. J. Slater. Fundamentals of Domination in Graphs. Marcel Dekker, Inc., 1998. 6. Luttrell, K. On the Neighbor-Component Order Connectivity Model of Graph Theoretic Networks. PhD thesis, Stevens Institute of Technology, 2013. 7. Mohan, J.J. and Kelkar, I. Restrained 2-domination number of complete grid graph. International Journal of Applied Mathematics and Computations, 4:352–358, 2012. 8. Ore, O. Theory of Graphs. American Mathematical Society Colloquium Publications, 1962. 9. Shaheen, R., Mahfud, S. and Almanea, K. On the 2-domination number of complete grid graphs. Open Journal of Discrete Mathematics, 7:32–50, 2017.

Graphic Approximation of Integer Sequences Brian Cloteaux

Abstract A variety of network modeling problems begin by generating a degree sequence drawn from a given probability distribution. If the randomly generated sequence is not graphic, we give two new approaches for generating a graphic approximation of the sequence. These schemes are fast, simple to implement, and only require a linear amount of memory. This allows approximation to be performed on very large integer sequences. Keywords Degree sequence · Graph generation · Graphic sequences MSC 68R10, 05C07

1 Introduction Creating models of real-world complex networks, such as social and biological interactions, remains an important area of research. The modeling and simulation of these systems often requires developing realistic graph models for the underlying networks embedded in these systems. A common first step in the creation of these graph models is to construct a graph with a given degree sequence. If we are able to construct a graph from an integer sequence, we say that the sequence is graphic. One problem in the random modeling of graphs is selecting a graphic integer sequence from some given probability distribution. More specifically, how do we

Official contribution of the National Institute of Standards and Technology; not subject to copyright in the United States. B. Cloteaux (□) National Institute of Standards and Technology, Applied and Computational Mathematics Division, Gaithersburg, MD, USA e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 S. Heuss et al. (eds.), Combinatorics, Graph Theory and Computing, Springer Proceedings in Mathematics & Statistics 462, https://doi.org/10.1007/978-3-031-62166-6_13

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deal with a randomly drawn integer sequence that is not graphic? For certain distributions, this can be a common occurrence. For instance, if we select two dominating nodes (nodes that are connected to every other node in the graph) and at least one node whose degree is one, we have an immediate contradiction if we try to construct a graph. This scenario often occurs when selecting from a truncated power-law distribution (where any selected value that is greater or equal to the sequence length is truncated to the length minus one). There have been two approaches to this problem. The first is to simply discard the sequence and repeatedly select a new sequence until a graphic sequence is found. This approach has been used in the past by the NetworkX [6] graph library. A disadvantage to this approach is that, as we just saw, for some distributions the probability of selecting a graphic sequence is very small. Thus the NetworkX library had placed an upper limit on the number of attempts to generate a graphic sequence before abandoning the task. This approach could lead to an expensive computational task that ultimately fails to produce any results. In response to this difficulty, a second approach, suggested by Mihail and Vishoi [9], is to find the closest graphic sequence to the original non-graphic degree sequence using some distance measure. Using a measure that they termed as the discrepancy, Mihail and Vishoi provided a polynomial-time algorithm for finding these graphic approximations by reducing this problem to a maximum cardinality matching problem; unfortunately, their algorithm is not practical for many cases. They posed the question whether a practical algorithm for determining nearest graphic sequence under the discrepancy measure could be found. This question was answered by Hell and Kirkpatrick [7] who introduced two algorithms for computing a minimal discrepancy graphic sequence in lower polynomial time. The Hell and Kirkpatrick approach involves constructing a non-random graph whose degree sequence has minimum discrepancy. For very large integer sequences, having to construct a realization in order to approximate the sequence can be problematic. Memory limitations potentially makes this type of approach expensive for extremely large instances. In this article, we consider two approaches to produce graphic approximations of integer sequences without having to construct a realization. The first approach gives a new algorithm to minimize the discrepancy metric. A second approach that we consider is to try to minimize the probability distribution distance between the sequences. Both approaches give fast algorithms that are easy to implement and only require a linear amount of memory.

2 Definitions and Needed Results Before introducing the algorithms, we need some preliminary definitions and results. A degree sequence .α = (α1 , α2 , ..., αn ) is a set of non-negative integers such that .α1 ≥ α2 ≥ ... ≥ αn ≥ 0. As a notational convenience, if a value a is represented r times in a sequence we represent that subsequence as .a r , i.e.,

Graphic Approximation of Integer Sequences

⎛ .

179

⎞

( ) ⎝a, ..., a ⎠ = a r . , ,, ,

(1)

r

If there exists a graph G whose node degrees match the sequence .α, then .α is said to be graphic; else, the sequence is called non-graphic. One assumption that we make in this article about any integer sequence .α is that .α has an even sum s and .s ≤ |α|(|α| − 1). We say that a sequence that meets these conditions is proper. It is straightforward to see that any sequence that is not proper cannot ever be graphic. Conversely, if this condition is met, then there will exist some sequence that has length .|α| and sum s and that is graphic (this follows from Chen [3], Lemma 1). As an observation, if a sequence only violates this condition by having an odd sum, we can modify the sequence by removing a random value in the sequence and replacing it with a new sampled value, repeating until the sequence’s sum becomes even. A tool we will use to work with degree sequences is the partial ordering called majorization. Majorization allows us to compare two degree sequences that have the same length and sum. We denote the set of all sequences having length n and sum s as .L(n, s). A degree sequence .α majorizes (or dominates) the degree sequence .β, denoted by .α ⪰ β, if for all k from 1 to n k Σ .

i=1

αi ≥

k Σ

βi ,

(2)

i=1

and if theΣ sums of theΣ two sequences are equal. If .α ⪰ β and there exists an index k where . ki=1 αi > ki=1 βi , then we say that .α strictly majorizes .β, denoted as .α ≻ β. If .α is a degree sequence and .αi ≥ αj + 2 where .i < j then the operation of subtracting one from .αi and adding one to .αj is called the unit transformation from i to j on .α. The new sequence created by this operation is denoted by .α[i → j ]. To designate the application of a sequence of unit transformations, S = ([i1 → j1 ], [i2 → j2 ], ..., [im → jm ]) ,

.

(3)

to a degree sequence, we use the notation .α[S]. We can recursively define this application as α[S] = (α[im → jm ]) [S ' ]

.

(4)

where .S ' = ([i1 → j1 ], [i2 → j2 ], . . . , [im−1 → jm−1 ]). When speaking about a sequence of unit transformations S, we will use the notation .|S| to denote the length of the sequence. Two sequences of unit transformations, such as .S1 and .S2 , can potentially / |S2 |. produce the same resulting sequence, i.e., .α[S1 ] = α[S2 ], even when .|S1 | =

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As an example, consider how .S1 = ([i → k]) and .S2 = ([i → j ], [j → k]) both will produce the same resulting sequence. Thus, we say that a sequence .S1 of unit transformations is minimal if there do not exist any sequences .S2 such that .α[S1 ] = α[S2 ] and .|S1 | ≥ |S2 |. There is a close relationship between majorization and unit transformations. Theorem 1 (Muirhead [10]) For any degree sequences .α and .β where .α ⪰ β, .β can be obtained from .α by a finite sequence of unit transformations. Over the set of integer partitions .L(n, s), majorization forms the lattice (L(n, s), ⪰ ) [2] where the meet operator is defined as follows:

.

(α ∧ β)k = min{ .

= min{

k Σ

αi ,

k Σ

i=1

i=1

k Σ

k Σ

i=1

αi ,

βi } −

k−1 Σ

(α ∧ β)i

i=1

βi } − min{

i=1

k−1 Σ

αi ,

i=1

k−1 Σ

(5) βi }.

i=1

For the sake of completeness, we also include the definition of the join operator. Although we will not reference it in this article, it is defined as: α∨β = .

Λ {φ : φ ⪰ α, φ ⪰ β}

= (α ∗ ∧ β ∗ )∗ ,

(6)

where .α ∗ is the conjugate sequence of .α. For our purposes, the usefulness of comparing degree sequences using majorization stems from the following result which shows that every sequence that is majorized by a graphic sequence must also be graphic. Theorem 2 (Ruch and Gutman [12], Theorem 1) If the degree sequence .α is graphic and .α ⪰ β, then .β is graphic. This result establishes the location of the graphic sequences in the lattice (L(n, s), ⪰ ). If the integer s is even, then there will be a small number of elements at the bottom of the partition lattice .L(n, s) that are graphic, while the remaining majority of the partitions will be non-graphic [11]. It also shows that the graphic sequences form a semi-lattice in .L(n, s).

.

3 Algorithm 1—Minimizing the Discrepancy We begin by examining Mihail and Vishoi’s original formulation for a sequence distance metric, the discrepancy. We introduce a new algorithm that for a given

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181

proper sequence finds a graphic sequence with minimal discrepancy using only a linear amount of memory.

3.1 Finding a Minimal Discrepancy Graphic Sequence Let .α be a non-graphic degree sequence and .β be a graphic degree sequence where α, β ∈ L(n, s) . Using the terminology of Mihail and Vishnoi [9], an index i is said to have a deficit of .αi − βi if .αi > βi . If .βi > αi , then the index i is said to have a surplus of .βi − αi . The discrepancy between the two sequences .α and .β, denoted by .Δ(α, β), is defined as the sum of the deficits and the surpluses over all the paired degrees,

.

Δ(α, β) =

n Σ

.

|αi − βi |.

(7)

i=1

To find a minimum discrepancy graphic approximation method, we will exploit the structure of the sequences within the majorization lattice .L(n, s). This requires that we first show several facts about the majorization lattice and the discrepancy. The first fact to note is that there is a direct relationship between the minimum path length in the lattice .L(n, s) to the minimization of discrepancies. We show this by reformulating a result originally by Arikati and Maheshwari. Theorem 3 (Arikati and Maheshwari, [1], Lemma 5.1) For the integer sequence α and for a minimal sequence of unit transformations S, then

.

Δ(α, α[S]) = 2 · |S|.

.

(8)

We can restate this result as when .α ⪰ β, then the discrepancy between the two sequences is twice the length of the shortest path between them in the majorization lattice. It also shows that for every minimal unit transformation set with the same length, say .|S1 | = |S2 |, all have identical discrepancy values from .α, i.e., .Δ(α, α[S1 ]) = Δ(α, α[S2 ]). Since the bottom element in .L(n, s) is graphic (again Chen [3], Lemma 1), it follows from Theorem 1 that an arbitrary sequence of unit transformations on a nongraphic sequence from that degree set will eventually produce a graphic sequence. From the last result, what we are interested in is finding a minimal sequence of unit transformations, i.e., a minimum length path in the lattice, to a graphic element. We start with a simple observation about how we can pick these unit transformations. Theorem 4 For degree sequences .α and .β in .L(n, s) where .α ⪰ β and for the unit transformations .[a → b] and .[c → d] where .c ≤ a < b ≤ d, .α[a → b] ⪰ β[c → d].

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Proof We observe that for the indices .k < c and .k > d, the majorization inequalities (2) are unchanged. Otherwise, there are two cases. Σ Σ Σ 1. If .c ≤ k < a or .b ≤ k < d, then . ki=1 βi [c → d] = ki=1 βi −1 < ki=1 αi = Σ k i=1 αi [a → b]. Σ k Σ k Σ 2. If .a ≤ k < b, then . ki=1 βi [c → d] = βi − 1 ≤ i=1 i=1 αi − 1 = Σ k i=1 αi [a → b]. ⨆ ⨅ Using this observation, we can immediately pick indices for the unit transformations which are guaranteed to preserve majorization. Now when we consider any minimal length sequence of unit transformations leading to a graphic sequence, we now have a constructive method to produce an alternative sequence of the same length. Theorem 5 For any degree sequence .α and sequence of unit transformations S, if α[S] is graphic, then for the sequence of unit transformations

.

⎞

⎛

⎟ ⎜ A = ⎝[1 → n], ..., [1 → n]⎠ ,, , ,

.

(9)

|S|

the degree sequence .α[A] is also graphic. Proof By inductively applying Theorem 4, we establish that .α[S] ⪰ α[A]. Then from Theorem 2, since .α[S] is graphic, .α[A] is also graphic. ⨆ ⨅ From Theorem 3, it follows that .Δ(α, α[S]) = Δ(α, α[A]). For a non-graphic sequence .α, this suggests a simple algorithm for finding a graphic sequence .β where .α ⪰ β and .Δ(α, β) is minimized. Starting with .α, we repeatedly apply the unit transformation .[1 → n] stopping when the derived sequence becomes graphic. There cannot exist a graphic sequence that .α majorizes with a smaller discrepancy. This procedure works for sequences that are comparable under the majorization operator. This raises the question of how to deal with sequences that have minimum discrepancy but are not comparable. For two sequences, .α and .β, to not be comparable under majorization either the sequences do notΣ have equal Σ Σ Σ psums p (. ni=1 αi /= ni=1 βi ) or there exist indices p and q such that . i=1 αi ≥ i=1 βi Σ q Σ q and . i=1 αi < i=1 βi . We show that if the non-graphic sequence is proper, then there will always be a comparable graphic sequence with minimum discrepancy. Let us first consider the problem of when the sums of the two sequences are not equal. Theorem 6 For .α ∈ L(n, s), there exists .β ∈ L(n, s) that is graphic and has the minimum discrepancy. Proof Take any graphic sequence not in .L(n, s) but having a minimum Σ .γ that isΣ discrepancy to .α. Assume that . ni=1 γi > ni=1 αi , otherwise, we would use the complement of the two sequences. Taking some realization G of .γ , then for any

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183

node i in G where .γi > αi , remove an arbitrary edge ij to create a new graph ' G γ ) < Δ(α, deg(G' )) causing a contradiction. Thus .2 + Σ n. If .γj > α' j , then Σ .Δ(α, n ' i=1 deg(G )i = i=1 γi while .Δ(α, γ ) = Δ(α, deg(G )). By iterating on this edge removal process, we derive a graphic sequence with minimal discrepancy and whose sum is equal to .α. ⨆ ⨅

.

This result shows that for a proper sequence .α ∈ L(n, s), that there must exist a graphic sequence .β ∈ L(n, s) that minimizes the discrepancy. Now, we take a step further to show that we can always find a graphic sequence that minimizes the discrepancy and is majorized by .α. Theorem 7 For .α, β ∈ L(n, s) Δ(α, β) ≥ Δ(α, α ∧ β)

.

(10)

Proof For each index k, the value .(α ∧ β)k can be one of four values. We consider each case to show that for all k, .|αk − βk | ≥ |αk − (α ∧ β)k |. Σ Σ k Σ k−1 Σ k 1. If . k−1 i=1 βi , then .(α ∧ β)k = αk and i=1 (α ∧ β)i = i=1 αi and . i=1 αi ≤ .αk − (α ∧ β)k = 0 Σ Σ k Σ k−1 Σ k 2. If . k−1 i=1 αi , then .(α ∧ β)k = βk and i=1 (α ∧ β)i = i=1 βi and . i=1 βi ≤ .αk − (α ∧ β)k = αk − βk . Σ Σ Σ Σ Σ 3. If . k−1 (α ∧ β)i = k−1 αi < k−1 βi and . ki=1 βi < ki=1 αi , then .(α ∧ i=1Σ i=1 i=1 Σ k βi − k−1 β) = i=1 i=1 αi . Using the inequalities, we have the bound .βk < Σ k−1 Σ kk β − α < < α − (α ∧ β)k < αk − βk . i=1 i i=1 i Σ αk and so .0 Σ Σ Σ k−1k Σ k−1 k−1 4. If . i=1 (α ∧ β)i = i=1 βi < i=1 αi and . ki=1 αi < ki=1 βi , then .(α ∧ Σ k Σ k−1 α − i=1 βi . Using the inequalities, we have the bound .αk < β) = i=1 Σ k−1i Σ kk i=1 αi − i=1 βi < βk and so .0 < (α∧β)k −αk < βk −αk or .|αk −(α∧β)k | < |αk − βk |. ⨆ ⨅ From Theorem 2, if .β is graphic, then .α ∧ β must also be graphic. Therefore, if the graphic sequence .β minimizes the discrepancy to .α, then from the above theorem, .Δ(α, β) = Δ(α, α ∧ β). Thus the consequence of these results is that for any non-graphic proper sequence there will always be a graphic sequence that minimizes discrepancy and is majorized by the non-graphic sequence. The algorithmic implication is that we only need to look at the sequences under the original sequence in the integer partition lattice when searching for a minimal discrepancy graphic sequence.

3.2 Creating a Fast Algorithm We now combine these results to form an approximation algorithm. The basic idea is to perform the unit transformation .[1 → n] on the sequence .α until it

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becomes graphic. This resulting graphic sequence is guaranteed to be have minimal discrepancy among all graphic sequences. In order to know when to stop the sequence modification, we use a form of the following classic test. Theorem 8 (Erd˝os and Gallai [5]) k Σ .

n Σ

αi ≤ k(k − 1) +

min{αi , k},

(11)

j =k+1

i=1

In our implementation, we use the following equivalent form of the Erd˝os-Gallai inequalities as derived by I. Zverovich and V. Zverovich [13, proof of Theorem 3]. k Σ .

⎛ αi ≤ k(n − 1) − k

i=1

k−1 Σ

Nα [i] −

i=0

k−1 Σ

⎞ iNα [i] ,

(12)

i=0

where .Nα [i] is the number of times the value i appears in the sequence .α. Combining this graphic testing result, we create Algorithm 1. If .φ is the minimum discrepancy to a graphic sequence, then the total number of iterations in the algorithm is .O(φ). A sequence that is already sorted can be tested whether √ it is graphic or not in time .O( s) were s is the sum of the sequence. By using the following algorithm, we consider an index in the sequence only until its Erd˝osGallai inequality is satisfied, and then we proceed to the next index. This idea is justified by the following result. Theorem 9 For the integer sequences .α and .β = α[1 → n], if .α satisfies the kth Erd˝os-Gallai inequality, then .β also satisfies the kth Erd˝os-Gallai inequality. Proof From .β = α[1 → n], then for .k ≥ 1 0≤

n Σ

.

n Σ

min{βi , k} −

i=k+1

min{αi , k} ≤ 1.

(13)

i=k+1

Then it follows from the majorization definition that k Σ i=1

βi =

k Σ

αi − 1 < k(k − 1) +

i=1

.

≤ k(k − 1) +

n Σ i=k+1

n Σ

min{αi , k} (14)

min{βi , k}

i=k+1

⨆ ⨅

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Now with careful algorithm design, the unit transformation step algorithm can √ be implemented in .O(1) time for a run time of .O(max{ s, φ}). The analysis of this algorithm still requires a lower-bound on the number of unit transformations required; it requires .Θ(n2 ) time in the worst case as shown in the following theorem. Algorithm 1 Algorithm for determining a graphic sequence with minimum discrepancy from a given non-graphic proper degree sequence Input: α - a potentially graphic sequence where α1 ≥ α2 ≥ ... ≥ αn ≥ 0, s - sum of sequence α Output: β - a graphic sequence with minimal discrepancy to α β←α dx ← max β, dn ← min β, Nβ ← binned distribution of β ds ← 0, js ← 0, ns ← 0 for k ← 1 to n do if β[k] < k + 1 then return β end if ds ← ds + β[k], ns ← ns + Nβ [k − 1], js ← js + (k − 1) · Nβ [k − 1] while ds > k(n − 1) − k · js + ns do β[Nβ [dx ]] ← β[Nβ [dx ]] − 1, β[n − Nβ [dn ] + 1] ← β[n − Nβ [dn ] + 1] + 1 js ← js − dn , ns ← ns − 1 if k − 1 ≥ dn + 1 then js ← js + dn + 1, ns ← ns + 1 end if if k ≥ Nβ [dx ] then ds ← ds − 1 end if Nβ [dx ] ← Nβ [dx ] − 1, Nβ [dx − 1] ← Nβ [dx − 1] + 1 Nβ [dn ] ← Nβ [dn ] − 1, Nβ [dn + 1] ← Nβ [dn + 1] + 1 if N [dx ] = 0 then dx ← dx − 1 end if if N [dn ] = 0 then dn ← dn + 1 end if end while end for return β

Theorem 10 In its worst case, Algorithm 1 requires .Θ(n2 ) unit transformations to transform into graphic sequence. Trivially, for the sequence .(s, 0, ..., 0) where .s = Θ(n2 ), it is obvious that .Θ(n2 ) unit transformations are needed to transform the value of the largest entry to .n − 1. At the same time, for any sequence, we can transform it into a near-regular sequence in .O(n2 ) steps. Even if we restrict the non-graphic sequences to those whose largest value is 2 .≤ n − 1, we still have a .O(n ) worst case number of unit transformations. For example, consider the family of sequences

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⎞

⎛

⎟ ⎜ α = ⎝n − 1, ..., n − 1, 0, ..., 0⎠ . ,, , , ,, , ,

(15)

.

n 2

n 2

For any sequence to be graphic, it must satisfy all the Erd˝os-Gallai inequalities. Consider the Erd˝os-Gallai inequality for the . n2 th index of .α, n

2 Σ .

i=1

αi =

n ⎞ Σ n ⎛ n n (n − 1) /≤ −1 + αi 2 2 2 n

(16)

i= 2 +1

Since this algorithm takes the largest value each time, we can rewrite the sequence .α after a sequence of .n/2 unit transformations as, ⎞

⎛ .

⎟ ⎜ ⎝c(n − 1), ..., c(n − 1), (1 − c)(n − 1), ..., (1 − c)(n − 1)⎠ , , ,, , , ,, , n 2

(17)

n 2

where .0 ≤ c ≤ 1. Thus we determine a lower bound on the number of unit transformations needed to become graphic by solving the following equation stemming from the Erd˝os-Gallai inequality, .

⎞ n ⎛ n n c(n − 1) = − 1 + (1 − c)(n − 1) 2 2 2

(18)

Solving for c shows that .c ≈ 14 , thus the number of unit transformations needed is n 1 2 .≈ 2 · 4 (n − 1) = Θ(n ). Practically, the number of unit transformations needed is usually much smaller than .O(n2 ). In Fig. 1, we see the average number of unit transformations needed to convert to a graphic sequence for a number of non-graphic integer sequences drawn from a power-law distribution. For these sequences, the number of unit transformations needed appears to have linear growth.

4 Algorithm 2—Minimizing the Probability Distribution Distance We now shift our perspective on the problem by examining the probability distribution distance between sequences instead of the discrepancy. For this change in the distance metric, we introduce a linear-time algorithm that produces a graphic sequence for a given proper sequence and, for many cases, approaches the minimal total variation distance of the probability distribution.

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Fig. 1 This figure represents the average number of unit transformation between a non-graphic sequence selected from a power-law distribution with exponent of 2 and its approximation created from Algorithm 1. Each point represents the average number of unit transformations for 30 sequences created at a given length. The error bars represent one standard deviation

4.1 An Approximation Scheme Under Total Variation Distance When Mihail and Vishoi originally suggested the discrepancy metric for measuring the distance between the original sequence and its approximation, they asked if there were other natural notions of distance between sequences, and for these other distance measures, are there efficient algorithms to graphicly approximate them [9, page 7, Q6]. For the problem of drawing a graphic sequence given a probability distribution, we propose that, often times, a more useful measure is to minimize the distance between the probability distributions of the two sequences. To define this measure, we need to rewrite a sequence as a discrete probability distribution. We start by writing the sequence .α = (1r1 , 2r2 , ..., (n − 1)rn−1 ) where all i where .0 ≤ ri ≤ n − 1. What we are interested in is finding a graphic sequence that minimizes the difference in its probability distribution compared to our original sequence. matching to the vector .(r1 , r2 , ..., rn−1 ) by the probability distribution of the approximation sequence. The probability of an individual value i being chosen is .P (α)i = ri /n, and we denote the probability distribution of a sequence .α as .P (α) = (r1 /n, r2 /n, ..., rn−1 /n). For measuring probability distances for creating approximations, we use the total variation distance. This distance measures the largest possible probability difference for an event between two distributions P and Q and is defined as

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T V (P , Q) = sup |Pω − Qω |.

.

(19)

ω∈Ω

4.2 A One-Pass Approximation Algorithm We present a new approximation scheme for the sequence .α by first introducing the graphic sequence T with the same length and sum as .α that has a number of desirable features. This sequence T is defined as follows: For the value .m = min{k|s ≤ k · (k − 1)}, ⎧ ⎪ (0n ) if s = 0, ⎪ ⎪ ) ⎪( ⎨ γ1 , ..., γm , 0n−m where γ = T (m, s) if n > m, .T (n, s) = ⎪ ⎪(n − 1, γ1 + 1, ..., γn−1 + 1) ⎪ ⎪ ⎩ where γ = T (n − 1, s − 2 · (n − 1)) if n ≤ m.

(20)

The first feature that we note is that the sequence T defines a threshold sequence (see [8], Theorem 1.2.4). A threshold sequence is a maximal graphic sequence in the majorization lattice (see [8], Theorem 3.2.2). In other words, there is no graphic sequence that majorizes the sequence .T (n, s). At the same time, we see from the definition that this sequence contains the minimum possible number of non-zero elements for a graphic sequence with a given sum s. Additionally, an advantage in using the sequence T is that we do not have to store or pre-compute this sequence. We can quickly compute any index of .T (n, s) in constant time as shown by the following result. ) ( Theorem 11 The sequence .T (n, s) = (p + 1)q , pp+1−q , q, 0n−p−2 where |√ p=

.

| 4s + 1 − 1 , 2

(21)

and q=

.

s − p(p + 1) . 2

(22)

Proof We show this result by induction. For the base step, if .s( = 2 then ) this definition produces the values .p = 1 and .q = 0 for the sequence . 12 , 0n−2 which matches the definition in Eq. (20). For the inductive step, we assume that for all sums less than s, the above result holds. Take the sequence .γ = T (n, s) where without a loss of generality we assume that .γn > 0. By reducing the sequence by removing the value .γ1 and subtracting one from the remaining values, we create the new sequence .γ ' = (γ2 − 1, ..., γn − 1) whose sum is .s ' = s − 2n. From the inductive hypothesis, this new sequence can

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189

⎞ ⎛ ' ' be written as .γ ' = (p' + 1)q , p'p +1−q , q ' where .p' = n − 2. By adding back ' .α1 along with adding one to the values in .γ , we can now write .γ as the value ( ) q p+1−q ' ' .γ = (p + 1) , p , q where .p = p + 1, .q = q + 1, and .s ' = s − 2(p + 1). We now consider the values for p and q in .γ . The value of the parameter q can be computed as follows:

s ' − p' (p' + 1) +1 2 . (23) s − p(p + 1) (s − 2(p + 1)) − (p − 1)p +1= . = 2 2 | |√ 4s+1−1 is equivalent to computing the integer p where The parameter .p = 2 .p(p + 1) ≤ s < (p + 1)(p + 2). Equation (23) establishes that .p(p + 1) ≤ s. The second part of the inequality holds as follows: q = q' + 1 =

s = s ' + 2(p + 1) = s ' + 2(p' + 2) = s ' + 2p' + 4 .

< s ' + 4p' + 6 = (p' + 1)(p' + 2) + 4p' + 6 '

(24)

'

= (p + 2)(p + 3) = (p + 1)(p + 2). Thus the premise is established.

⨆ ⨅

We apply this result to create the approximation algorithm. The idea is to take the meet operation between the original non-graphic sequence .α and the sequence T , where the length and sum of .α and T are equal. Since .α∧T ⪯ T , then by Theorem 2 the resulting sequence will be graphic. Also, since we are able to compute each value of T in constant time, we only need one pass through the sequence .α to create the graphic approximation for a run-time of .O(n). The full algorithm is shown in Algorithm 2. We point out a couple of practical points about this algorithm. While a degree sequence is being generated, it can be sorted simultaneously by using a binned sort. The√sorted sequence can be tested whether it is graphic or not in an additional .O( s) steps [4]. If the sequence is not graphic, this approximation scheme only requires one more pass through the sequence to produce a graphic sequence.

4.3 Comparing the Resulting Sequences What is advantage of this approach? For the general case of this problem, the time needed to compute a minimum total variation graphic sequence for a given nongraphic sequence is an open problem. We show that for many useful distributions, the distribution arising from the approximation sequence created by Algorithm 2 will asymptotically approach a minimal total distance as the length of the sequence

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Algorithm 2 Graphic sequence approximation algorithm Input: α - a potentially graphic sequence where α1 ≥ α2 ≥ ... ≥ αn ≥ 0, s - sum of sequence α Output:| β - a graphic | sequence √ 4s+1−1 p← 2 q ← s−p(p+1) ( 2) β ← 0|α| , as ← 0, bs ← 0, ts ← 0 for i ← 1 to |α| do as ← as + α[i] ti ← 0 if i ≤ q then ti ← p + 1 else if i > q & i ≤ p + 1 then ti ← p else if i = p + 2 then ti ← q end if ts ← ts + ti β[i] ← min{as , ts } − bs bs ← bs + β[i] end for return β

increases. In particular, this applies to sequences where the sum s of the sequence √ is . s = o(n). These sequences include many commonly used distributions, such as power-law distributions. Theorem 12 Let .F(n, s) be a family of integer sequences where for any .α ∈ F(n, s) whose sum is s and length is n, then .s = o(n2 ). For .α ∈ F(n, s), .

lim T V (P (α), P (α ∧ T (n, s))) = 0.

n→∞

(25)

Proof From the definition of the sequence T , the number of non-zero entries it contains is m. From the definition of the meet operation, the values for the sequence .T ∧ α will be equal to .αi for all indices .i > m. It follows from Theorem 11, that a √ simple upper bound on m is . s. Thus an upper bound on the possible change in the probability of any one value in the distribution is √ s m < . . sup |P (α)ω − P (α ∧ T (n, s))ω | ≤ n n ω

(26)

Since .s = o(n2 ), then it follows √ s . lim T V (P (α), P (α ∧ T (n, s))) ≤ lim = 0, n→∞ n→∞ n completing the argument.

(27) ⨆ ⨅

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191

Fig. 2 This figure represents the average total variation distance between a non-graphic sequence selected from a power-law distribution with exponent of 2 and its approximation created from Algorithm 2. Each point represents the average distance for 30 sequences created at a given length. The error bars represent one standard deviation

This convergent behavior is shown in Fig. 2. In this example, for each point, 30 non-graphic sequences with even sum where s is drawn from a power-law distribution with exponent 2 and the average total variation distance to their approximation was computed. We see that the average total variation distance between the original sequences and their approximations approaches zero as the sequence length grows.

5 Conclusions We give two simple and fast methods to approximate non-graphic sequences with graphic ones. These approaches are particularly well-suited for large sequences, since they do not require creating realizations, but instead simply require storing the sequence. We also introduce the idea that a more practical distance metric might be the probability distance between two sequences, rather than their discrepancy. Future research directions include looking at other possible measures, such as the Jensen-Shannon divergence. In addition, we plan to release these algorithms to the network research community for use in modeling applications.

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References 1. Arikati, S.R., Maheshwari, A.: Realizing degree sequences in parallel. SIAM J. Discrete Math. 9(2), 317–338 (1996). https://doi.org/10.1137/S0895480194267932 2. Brylawski, T.: The lattice of integer partitions. Discrete Math. 6, 201–219 (1973). https://doi. org/10.1016/0012-365X(73)90094-0 3. Chen, Y.C.: A short proof of Kundu’s k-factor theorem. Discrete Math. 71(2), 177–179 (1988). http://dx.doi.org/10.1016/0012-365X(88)90070-2 4. Cloteaux, B.: Is this for real? Fast graphicality testing. Computing in Science & Engineering 17(6), 91–95 (2015). https://doi.org/10.1109/MCSE.2015.125 5. Erd˝os, P., Gallai, T.: Graphs with prescribed degrees of vertices. Mat. Lapok 11, 264–274 (1960), Hungarian 6. Hagberg, A.A., Schult, D.A., Swart, P.J.: Exploring network structure, dynamics, and function using NetworkX. In: Proceedings of the 7th Python in Science Conference (SciPy2008). pp. 11–15. Pasadena, CA USA (Aug 2008) 7. Hell, P., Kirkpatrick, D.: Linear-time certifying algorithms for near-graphical sequences. Discrete Math. 309(18), 5703–5713 (2009). https://doi.org/10.1016/j.disc.2008.05.005 8. Mahadev, N., Peled, U.: Threshold Graphs and Related Topics. North Holland (1995), Annals of Discrete Mathematics 56 9. Mihail, M., Vishnoi, N.: On generating graphs with prescribed degree sequences for complex network modeling applications. In: Proceedings of the 3rd Workshop on Approximation and Randomization Algorithms in Communication Networks (ARACNE). pp. 1–11 (2002) 10. Muirhead, R.F.: Some methods applicable to identities and inequalities of symmetric algebraic functions of n letters. Proceedings of the Edinburgh Mathematical Society 21, 144–157 (1903) 11. Pittel, B.: Confirming two conjectures about the integer partitions. Journal of Combinatorial Theory, Series A 88(1), 123–135 (Oct 1999). https://doi.org/10.1006/jcta.1999.2986 12. Ruch, E., Gutman, I.: The branching extent of graphs. Journal of Combinatorics, Information, & System Sciences 4(4), 285–295 (1979) 13. Zverovich, I., Zverovich, V.: Contributions to the theory of graphic sequences. Discrete Mathematics 105(1-3), 293–303 (1992). https://doi.org/10.1016/0012-365X(92)90152-6

Changing the Uniform Spectrum by Deleting Edges Drake Olejniczak and Robert Vandell

Abstract A graph is said to be k-uniformly connected if there exists a path of length k between each pair of vertices. This generalizes the well-known concept of a Hamiltonian-connected graph—a graph, order n, in which there exits a Hamiltonian path (path of length .n − 1) between each pair of vertices. That is, a graph is Hamiltonian-connected if and only if it is .(n − 1)-uniformly connected. One can also say a graph is complete if and only if it is 1-uniformly connected. The uniform spectrum of a graph G is the set of all k for which G is k-uniformly connected. In this chapter, we investigate the impact of adding or deleting vertices or edges on the uniform spectrum of a graph. Some general results are presented as well as analyses of specific classes of graphs such as bipartite graphs and wheels. Keywords Connectedness · Hamiltonicity

1 Introduction When using graphs to model networks, one of the fundamental properties that we are interested in is connectivity, which has been studied in various forms for many years. We know that in a graph G of diameter d, every pair of vertices are connected by a path of length at most d. What else can we say about the lengths of paths between vertices in a connected graph? Let G be a connected graph of order .n ≥ 2 with k an integer such that .1 ≤ k .≤ n − 1, then we say that G is k-uniformly connected if every pair of distinct vertices in G are connected by a path of length k. For example .K4 –e is 2-uniformly connected, but not 1- or 3-uniformly connected. A graph G can only be k-uniformly connected if k is at least the diameter of G. For a graph of order n, .(n−1)-uniformly connected is equivalent to Hamiltonian-connected. The set of all positive integers k

D. Olejniczak (□ ) · R. Vandell Department of Mathematics, Purdue University Fort Wayne, Fort Wayne, IN, USA e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 S. Heuss et al. (eds.), Combinatorics, Graph Theory and Computing, Springer Proceedings in Mathematics & Statistics 462, https://doi.org/10.1007/978-3-031-62166-6_14

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for which a graph G is k-uniformly connected is called the Uniform Spectrum of G and is denoted .U S(G). For example, .U S(K4 −e) = {2}, while .U S(K4 ) = {1, 2, 3}. From the example above, we can see that the addition or deletion of a single edge can impact the Uniform Spectrum of a graph. One can look at this phenomena from two perspectives. First, removing a single edge from .K4 eliminates two numbers from .U S(K4 ). The other perspective is that adding an edge to .K4 − e adds two numbers to .U S(K4 − e). In this chapter we begin to look at how removing edge(s) from a graph impacts its uniform spectrum. To investigate this question, a natural place to start would be classes of graphs with extreme uniform spectra. The complete graphs .Kn with .n ≥ 2 are the only graphs which have a full spectrum .{1, 2, 3, . . . , n − 1}. In Sect. 2, we consider the effect of removing a spanning cycle from .Kn . Of course, if a graph has a pair of nonadjacent vertices, then 1 is no longer in the Uniform Spectrum. We will say a graph G of order n has .nearlyf ullspectrum if 1 is the only value missing (.U S(G) = {2, 3, 4, . . . , n − 1}). In [2], the following proposition is established. Proposition 1 For each integer .n ≥ 6, the smallest size of a k-uniformly connected graph of order n and maximum degree .n − 1 for every integer k with .2 ≤ k ≤ n − 1 is .2n − 2. Furthermore, the wheel .Wn of order n is the only graph of order n and smallest size that is a k-uniformly connected graph of order n for each integer k with .2 ≤ k ≤ n − 1. In other words, every graph of order n with nearly-full spectrum has size at least 2n − 2, and the wheel graph .Wn is the unique graph which achieves this minimum. So, .Wn is an example of an extreme class of graphs whose uniform spectrum must change after removing a set of edges. Hence, the majority of this paper is devoted to determining .U S(Wn − X) where X is a subset of the edges of .Wn . We call an edge of .Wn a tire edge if it is an edge on the outside cycle of .Wn , and the edges joining the central vertex to the outside cycle are called spokes. In Sect. 3, we establish a lemma that is used in Sect. 4 to prove the general result for .U S(Wn − X) where X is any set of tire edges. In Sect. 5, we determine the uniform spectrum of .Wn after removing one of the spokes.

.

2 Complete Graphs When n is small, the uniform spectrum of .Kn is sensitive to the removal of only a few edges. For larger n, one can remove a spanning cycle from .Kn and still have a nearly-full spectrum. For example, .U S(K2 ) = {1}, hence .U S(K2 − e) = ∅. For .n = 3 or .n = 4, the removal of a single edge from .Kn will impact the upper end of the spectrum as well. For .3 ≤ n ≤ 4, let .V (Kn ) = {v1 , v2 , . . . vn }. Suppose .e = v1 v2 is deleted. When .n = 3, there is no longer a .v2 − v3 path of length 2, so .U S(K3 − e) = ∅. For .n = 4, there is no longer a .v3 − v4 path of length 3. There are still .u − v paths of length 2 between any pair of vertices. Hence, .U S(K4 − e) = {2}.

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The removal of a single edge from .Kn , .n ≥ 5, does not impact the upper end of the spectrum. In [1], the following result shows that for n large enough, we can remove many more edges without impacting the Uniform Spectrum further. Theorem 1 If G is a graph of order .n ≥ 4 with .δ(G) ≥ n+2 2 , then .U S(G) = [1, n − 1] if G is complete and .U S(G) = [2, n − 1] if G is noncomplete. An implication of this is that for .n ≥ 8, .U S(Kn − Cn ) = [2, n − 1]. When .n = 3 or 4 then .(Kn − Cn ) is disconnected, so .U S(Kn − Cn ) = ∅. .(K5 − C5 ) = C5 , so .2 ∈ / U S(K5 − C5 ). Also there are no length 2 paths between antipodal points in .U S(K6 − C6 ), so .2 ∈ / U S(K6 − C6 ). Lemma 1 .U S(K7 − C7 ) = [2, 6] If we let .V (K7 ) = {v0 , v1 , . . . , v6 }, then .v0 v4 v1 , v0 v2 v4 v1 , v0 v2 v4 v6 v1 , .v0 v5 v2 v4 v6 v1 , v0 v5 v2 v4 v6 v3 v1 are .v0 v1 -paths of lengths 2–6. .v0 v4 v2 , .v0 v4 v6 v2 , .v0 v4 v1 v6 v2 , .v0 v4 v1 v3 v6 v2 , .v0 v4 v1 v5 v3 v6 v2 are .v0 v2 -paths of lengths 2–6. .v0 v5 v3 , .v0 v2 v5 v3 , .v0 v2 v5 v1 v3 , .v0 v4 v2 v5 v1 v3 , .v0 v4 v2 v5 v1 v6 v3 are .v0 v3 -paths of lengths 2–6. By symmetry, .U S(K7 − C7 ) = [2, 6]. Theorem 2 .U S(Kn − Cn ) = [2, n − 1] iff .n ≥ 7. This means that for .n ≥ 7 we can remove any collection of disjoint paths, and the remaining graph would still have a nearly-full spectrum. From here, one may have to remove many additional edges from .Kn in order to reduce the spectrum. Instead, we investigate wheels as these graphs have nearlyfull spectrum and are extreme in the sense that removing any edge must reduce the spectrum.

3 The Uniform Spectrum of G + K1 Since removing a single edge from the cycle of a wheel .Wn = .Cn−1 + K1 leaves us with a fan .Pn−1 + K1 a natural starting place would be to determine the uniform spectrum of fans. The result developed in this section can be applied more generally to graphs of the form .G ∼ = H + K1 were H is any graph, where .H + K1 is the join of H and .K1 obtained by adding a vertex v to H along with all edges of the form uv where .u ∈ V (H ). As we will see, the shortest “long” path in H characterizes the uniform spectrum of these class of graphs. Let u and v be vertices in a graph G. A .u − v geodesic is a shortest .u − v path, and the length of a .u − v geodesic is the distance between u and v and is commonly denoted .d(u, v). The eccentricity of a vertex u, denoted .e(u), is the distance from u to a vertex farthest from u. The radius of a graph G, denoted .rad(G), is the minimum eccentricity over all vertices of G. While .u−v geodesics give rise to these well-studied and commonly used graph metrics, the lesser-known concept of a .u−v detour, introduced by Chartrand, Zhang, and Haynes in [3], plays an important role in determining .U S(H + K1 ).

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A .u − v detour is a longest .u − v path in G and the length of a .u − v detour is the detour distance .D(u, v) between u and v. The detour eccentricity of a vertex u, .eD (u) is the detour distance from u to a vertex farthest from u, i.e., .eD (u) = max{D(u, v) : v ∈ V (G), v /= u}. The detour radius of a connected graph, .radD (G) is the minimum detour eccentricity over .V (G), i.e., .radD (G) = min{eD (u) : u ∈ V (G)}. Lemma 2 Let G be a connected graph of order .n ≥ 2 and let .G∗ = G + K1 . Then ⎧ U S(G∗ ) =

.

[2, radD (G) + 1] if G is not complete [1, n] if G is complete

Proof If .G = Kn , then .G∗ = Kn+1 and .U S(G∗ ) = [1, n]. So, we may assume that G is not complete. Let v be the vertex joined to G in .G∗ and let .x, y ∈ V (G∗ ). We show there is an .x − y path of length k for each .2 ≤ k ≤ radD (G) + 1 and every pair of vertices .x, y. First, since v is adjacent to every pair of vertices in G it follows that there is an .x − y path of length 2. Since G is connected and order at least 2, x is adjacent to some other vertex in G and there is an .x − v path of length 2. So, we may assume that .k ≥ 3. We consider two cases. Case 1 Assume .y = v. We have that .eD (x) ≥ radD (G) ≥ k − 1. So, there exists a vertex .x ' ∈ V (G) such that .D(x, x ' ) ≥ k − 1. Hence, there is an .x − x ' path P of length at least .k − 1 in G. We can then choose a vertex .x '' such that .P ' is an .x − x '' subpath of P of length exactly .k − 1 in G. Then .P ' + x '' v is an .x − y path of length k in .G∗ . Case 2 Assume .x, y /= v. By case 1, there is an .x − v path P in .G∗ of length k for each .3 ≤ k ≤ radD (G) + 1. Suppose .P = (x = x1 , x2 , · · · xk , v). If .y ∈ / P, then .P ' = (x = x1 , x2 , · · · xk−1 , v, y) is an .x − y path of length k. So, assume ' .y = xi for some .2 ≤ i ≤ k − 1. Then .P − y consists of the two paths .P : (x = '' x1 , · · · xk−1 ) and .P : (v, xk , xk−1 , · · · xi+1 ). An .x − y path of length k in .G∗ is obtained by appending .P ' with the edge .xk−1 v, appending .P '' with the edge .xi+1 y, and concatenating the results. So, .[2, radD (G) + 1] ⊆ US(G∗ )]. Since there exists a pair of nonadjacent vertices in .V (G∗ ), .1 ∈ / US(G∗ ). It remains to be seen that if .k ≥ radD (G) + 2, then there is some pair of vertices x and y with no .x − y path of length k. Let .k ≥ radD (G) + 2 and assume that for any two vertices x and y there is an ∗ .x − y path of length k in .G . In particular, there is an .x − v path of length k. Let ' ' .x be the vertex on this path adjacent to v. Then there is a .x − x path of length ' .k − 1 ≥ radD (G) + 1 entirely contained in G. Hence, .D(x, x ) ≥ radD (G) + 1, so .eD (x) ≥ radD (G) + 1 for every vertex .x ∈ V (G). However, this implies that .radD (G) ≥ radD (G) + 1, a contradiction. ⨅ ⨆

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Proposition 2 Let .F1,n be the fan graph of order ┌ ┐.n + 1 ┐ ┐obtained by joining .K1 to the path .Pn of order .n ≥ 3. Then .U S(F1,n ) = 2,

n+1 2

.

Proof Since .F1,n = Pn + K1 = Pn∗ and since .Pn is connected and noncomplete for .n ≥ 3, it follows from Lemma 2 that .U S(F1,n ) = [2, radD (Pn )]. So, we need only determine the detour radius of .Pn . Let .Pn = (v1 , v ┐2 , · · · ┐, vn ), then .eD (vi ) = for each .1 ≤ i ≤ n. max{i − 1, n − i}. By the pigeonhole principle .eD (vi ) ≥ n−1 2 ┐ ┐ ┐ ┐ ┐ ┐ n−1 n−1 Further, let .k = n+1 , then .eD (vk ) = . Hence, .radD (Pn ) = . The 2 2 2 result then follows. ⨅ ⨆ Lemma 2 can also be used to establish the uniform spectrum of the wheel graph of order n which is obtained by joining .K1 to a cycle of order .n − 1. That is, .Wn = Cn−1 + K1 . Example 1 The uniform spectrum of the wheel graph of order .n ≥ 5 is given by U S(Wn ) = [2, n − 1].

.

Proof Let v be a vertex in .Cn−1 , then .eD (v) = n − 2. Hence, .radD (Cn−1 ) = n − 2. Now, .Wn = Cn−1 + K1 and .Cn−1 is both connected and noncomplete for .n ≥ 4. So, by Theorem 2, it follows that U S(Wn ) = [2, radD (Cn−1 ) + 1] = [2, n − 1]

.

⨅ ⨆

.

Lemma 2 cannot, however, be used directly to determine the uniform spectrum of, say, the friendship graphs, which can be described as the join of .K1 to a collection of disjoint edges. More importantly, the removal of a set of rim edges from the wheel can also be described similarly as the join of .K1 to a disjoint collection of paths. So, we extend Lemma 2 to include the case when G is disconnected. Lemma 3 Let G be a noncomplete graph of order .n ≥ 2 with connected components .G1 , G2 , · · · Gt and let .r = min{radD (Gi )}t1 . Let .G∗ = K1 + G, then if .r ≥ 1, US(G∗ ) = [2, r + 1]

.

If .r = 0, then .U S(G∗ ) = ∅. Proof If .r = 0, then .Gi ∼ = K1 for some i, say .G1 ∼ = K1 . Let .u ∈ V (G1 ) and observe that there is only one .u − v path and its length is 1. Since G is not complete, ∗ .U S(G ) = ∅. Now, let .x, y ∈ V (G∗ ) and let v be the vertex joined to G, we first show there is an .x − y path of length k for each .2 ≤ k ≤ r + 1. If .x, y ∈ V (Gi ) for some i, or if one of x or y is v, then .x, y ∈ V (G∗i ) where .G∗i = v + Gi . By Lemma 2, there is an .x, y path of length k for each .2 ≤ k ≤ radD (Gi ) + 1. Hence, there is an .x − y path of length k for each .2 ≤ k ≤ r + 1. So, we may suppose .x ∈ V (Gi ) and .y ∈ V (Gj )

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for some .i /= j . Since .x, v ∈ V (G∗i ), there is an .x − v path P of length .k − 1 for each .2 ≤ k ≤ r + 1. Then .P + vy is an .x − v path of length k in .G∗ . Now suppose that for every pair .x, y ∈ V (G∗ ) there exists an .x −y path of length k in .G∗ for some .k ≥ r + 2. Without loss of generality, assume .radD (G1 ) = r. Let .x ∈ V (G1 ), then there is an .x − v path of length k in .G∗ . Since the removal of v disconnects the graph .G∗ , it follows that P is entirely contained in .G∗1 . Let ' ' ' .x ∈ V (G1 ) be the vertex adjacent to v on P , then .P −x v is an .x −x path of length .k− in .G1 . Since this is true for each vertex .x ∈ V (G1 ), .radD (G1 ) ≥ k − 1 ≥ r + 1. However, .radD (G1 ) = r. ⨅ ⨆

4 Wn Minus Rim Edges We can now easily obtain the uniform spectrum of .Wn − X where X is any set of edges on the outside cycle. Theorem 3 Let .X ⊆ E(Wn ) be a nonempty subset of edges on the outside cycle of Wn for .n ≥ 4, let v be the central vertex of .Wn , let .G = Wn − X, and let l be the smallest order of the components of .G − v. Then

.

⎧ ⎨∅ if l = 1 ┌ ┐ ┐ ┐ .U S(G) = ⎩ 2, l+1 if l > 1 2 Proof Since X is nonempty, each component of .G − v is a path. Hence, G is obtained by joining .K1 to a disjoint collection of paths. If .l = 1 or, alternatively, X contains two adjacent edges, then some component of .G − v is trivial. Since .n ≥ 4, and by Lemma 3, it follows that .U S(G) = ∅. Hence, we may assume that .l ≥ 2. ┐ ┐ In the proof of Proposition 2, it was shown that .radD (Pt ) = t−1 2 . The result then follows by Lemma 3. ⨅ ⨆ UThe friendship graph of odd order n, .Fn , can be described by .Fn = K1 + ˜ ( ti=1 K2 ) where .t = n−1 2 . The semi-friendship graph of even order .n ≥ 6, .Fn , is defined similarly, but where one of the edges is replaced with the path .P3 (Fig. 1). While it was shown in [1] that .2 ∈ U S(Fn ) and .2 ∈ U S(F˜n ) and that these graphs are unique graphs that have minimum size over all graphs G of order n and .2 ∈ U S(G), .U S(Fn ) and .U S(F˜n ) were not fully determined. Lemma 3 can be used to establish these facts. Alternatively, for odd n, the friendship graphs can be obtained from .Wn by removing a perfect matching of the outside cycle, i.e., every other tire edge is removed. For even n, the semi-friendship graphs can be obtained by removing a maximal matching of the outside cycle. With this description, .U S(Fn ) and .U S(F˜n ) follow immediately from Theorem 3.

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Fig. 1 Friendship graphs and semi-friendship graphs

Corollary 1 The uniform spectrum of the friendship graph of odd order .n ≥ 5 and the semi-friendship graph of even order .n ≥ 6 each have the singleton uniform spectrum .{2}. Proof Let v be the central vertex of .Wn for .n ≥ 5. For odd n, removing a perfect matching of the outside cycle, X, from .Wn yields the friendship graph of odd order n. Then .Wn − v − X consists of . n−1 2 disjoint edges, that is, paths of order 2. By Theorem 3, .U S(Wn − X) = U S(F = {2}. When n is even, a maximal matching n | )| = n2 −1 tire edges. .Wn −v−X then consists X of the outside cycle consists of . n−1 2 of . n2 − 2 disjoint edges and one .P3 , disjoint from the other edges. Since .n ≥ 6, there is at least one of these edges present in .Wn − X, and Theorem 3 guarantees that .U S(Wn − X) = U S(F˜n ) = {2}. ⨅ ⨆

5 Wheel Minus Spokes Let X be a set of spokes in .Wn , then the maximum degree of .Wn − X is .n − 1 − |X|. So, the problem of determining .U S(Wn − X) when X is a set of spokes is fundamentally different than when X is a set of tire edges. In particular, Lemma 3 cannot be directly applied to this case. So, a new approach must be taken. The difficulty of the problem is demonstrated in the complexity of the argument used in determining .U S(Wn − e) when e is one of the spokes of .Wn and in the somewhat surprising result obtained for the uniform spectrum of this graph. Throughout this section, label the vertices of .Wn = K1 + Cn−1 by letting .Cn−1 = (u0 , u1 , · · · un−2 , u0 ) be the outside cycle and v the central vertex where the subscripts of the outside cycle are expressed modulo .n − 1. Theorem 4 Let .Wn be as described above for .n ≥ 5, and let e be one of the spokes of .Wn , i.e., .e = ui v for some .0 ≤ i ≤ n − 2. Then U S(Wn − e) =

.

⎧ [3, n − 2] if n is even n+3 [3, n−1 2 ] ∪ [ 2 , n − 2] if n is odd

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D. Olejniczak and R. Vandell

Proof Without loss of generality, assume .e = u0 v. Observe first that the forward inclusion holds. Indeed, – there is no .u0 − u1 path of length 2 – Since removing the vertices .{u1 , un−2 } disconnects .Wn − e, .Wn − e is not 3connected and is therefore not Hamiltonian-connected. In particular, there is no .u1 − un−2 path of length .n − 1. When n is odd, we also have that – there is no .v − u n+1 path of length . n+1 2 . 2

To establish the reverse inclusion, we show there is an .x − y path of length k for each .3 ≤ k ≤ n − 2 when n is even and an .x − y path of length k for .3 ≤ k ≤ n−1 2 and . n+3 ≤ k ≤ n − 2 when n is odd and for all .x, y ∈ V (Wn ). 2 Case 1: .x = ui , .y = uj for some .1 ≤ i < j ≤ n − 2. Let .d = j − i ≥ 1. We first consider the subcase when .d = 1. Subcase 1a: .d = 1. Since .U S(Wn ) = [2, n − 1], let P be a .ui − uj path of length k in .Wn for some .3 ≤ k ≤ n − 2. If .v ∈ / V (P ), then .k = n − 1, so we may assume that .v ∈ V (P ). Then P has the form .(ui , ui−1 , · · · us , v, ut , ut−1 , · · · uj +1 , uj ) for some s and t, where possibly .s = i and/or .t = j . If neither s nor t is equal to 0, then P is also a path in .Wn − e of length k and we are done. So, assume .s = 0 or .t = 0. Since we may relabel .ui as .uj and vice versa, we may assume, without loss of generality, that .s = 0 and .j ≤ t ≤ n − 2 (Fig. 2). We now apply one of two operations to P to obtain .P ' or .P '' : – .P ' = P − {u0 v, u0 u1 , ut v} + {u1 v, ut+1 v, ut+1 ut } – .P '' = P − {u0 v, ut v, ut−1 ut } + {un−2 v, u0 un−2 , ut−1 v} Clearly, .P ' is a .ui − uj path of length k in each case. If at least one of these two operations is available and produces a path that does not contain the edge .u0 v, then .P ' is a path of length k in .Wn − e and we are done. So, assume that both of these operations do not produce a desired path, i.e., a well-defined path that does not contain .u0 v. Since .s = 0 and .i /= 0, the edge .u1 v appearing in .P ' is always available, even when .i = 1. So, the only obstruction for .P ' to have the desired property is if .t + 1 = 0 and it follows that .t = n − 2. In this case, .P '' produces a

Fig. 2 Case 1a in Theorem 5.1

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desired path except if .t − 1 = n − 3 = j . However, since .d = 1, .i = n − 4 = 0 which implies .n = 4 < 5, a contradiction. Subcase 1b: .d ≥ 2. In this case, consider the path .Q = (ui , ui+1 , · · · us , v) and .R = (v, ut , ut−1 , · · · uj ) for .i ≤ s ≤ j − 1. Since .s /= 0, the concatenation .P = Q + R is a .ui − uj path of length k so long as .t /= 0. So, assume .t = 0. We again describe two possible operations on both Q and R to produce .Q' and .R ' and show that one of these operations produces a desired path .P ' = Q' + R ' or '' '' '' .P = Q + R of length k not containing .u0 v. – .Q' = Q − {us v, us us−1 } + us−1 v and .R ' = R − vut + {vut+1 , ut+1 ut }. – .Q'' = Q − us v + {us+1 v, us us+1 } and .R '' = R − {vut , ut ut−1 } + vut−1 The path .Q' is well-defined except when .s = i. The path .R ' is well-defined except when .t = i − 1. If .t = i − 1 and .s = i, then .d = 1, and we are done. If '' '' .t = i − 1, then .i = 1. If .d ≥ 2, then .Q is well-defined and .R is well-defined except when .t = j + 1. However, we then have that .i − 1 = t = 0 = j + 1 and .j = n − 2. In this case, we can relabel .ui to .uj and vice-versa and obtain the case when .d = 1, so we are done. Case 2: .x = ui for some .1 ≤ i ≤ n − 1 and .y = v Let P be a .ui − v path of length k for some .3 ≤ k ≤ n − 2 in .Wn . If .i = 0, then since the length P is at least 3, P necessarily avoids the edge .u0 v and we are done. So, we may assume .i /= 0. Then P has the form .(ui , ui+1 , · · · us , v) where .s = k + i − 1, or P has the form .(ui , ui−1 , · · · ut , v) where .t = i − k + 1 (Fig. 3). One of these two paths is a .ui − uj path of length k in .Wn − e except in the case when .s = k + i − 1 = 0 and .t = i − k + 1 = 0. In this case .k + i − 1 ≡ i − k + 1 (mod (n − 1)). Hence, n+1 .k + i − 1 = n + i − k + 1 which implies .k = 2 and n is odd. In any case, we have produced an .x − y path of length k in .Wn − e for each .x, y ∈ V (Wn ). ⨅ ⨆ Fig. 3 Case 2 in Theorem 5.1

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Although the method used in the proof of Theorem 4 cannot be easily generalized to deal with the case of removing an arbitrary set of spokes, it can be extended to the case when X is a set of consecutive spokes to prove Theorem 5. The proof has more cases, hence is longer and more tedious. We present the result here without including the proof. Theorem 5 Let X be a set of r consecutive spokes of .Wn for .n ≥ 5, i.e., .X = {ui v, ui+1 v, · · · ur v}. Then ┐ ┐ ┌ | | ┐ ┌ ┌ n+r n−r ∪ + 1, n − 1 − r .U S(Wn − X) = 2 + r, 2 2

6 Problems and Future Work In Theorem 3, we can see that removing two edges from .Wn results in a graph with empty uniform spectrum. Combining Theorems 3 and 4, we can see that removing any one edge from .Wn will not result in a graph with empty uniform spectrum. Hence, the minimum number of edges whose removal from .Wn , .n ≥ 5, results in a graph with empty uniform spectrum is 2. This suggests the following problem. Problem 1 Given a graph G, what is the smallest number of edges whose removal results in a graph with empty uniform spectrum? If G is a graph with minimum degree .δ(G), then removing .δ(G) − 1 edges incident with the vertex of minimum degree with result in a graph with minimum degree 1. Any graph of order .n ≥ 3 with .δ(G) = 1 has empty uniform spectrum since there is only one path between this vertex of smallest degree and its neighbor and the length of this path is 1. However, there are sets X of edges of .Wn with .|X| = 2 whose removal does not result in an empty uniform spectrum. Problem 2 Let G be a graph of order n and let .X ⊆ E(G). Determine the minimum value k such that if .|X| ≥ k, then .U S(G − X) = ∅. In both the complete graphs and the Wheel graphs, we saw that removing any edge reduces the spectrum. However, for .n ≥ 7, consider .Kn − e whose uniform spectrum is .[2, n − 1]. In this case, we saw that we would have to remove at least n more edges in order to reduce the spectrum further and this is likely a weak lower bound. This suggests the following. Problem 3 Let G be a graph of order n. What is the most number of edges that must be removed in order to result in a graph .G' with .U S(G' ) ⊂ U S(G). The technique used in the proof of Theorem 4 leaves much to be desired in the way of a succinct proof of this result. Moreover, it would be desirable to produce a

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proof for Theorem 5 that is short enough to be included in this paper. Even further, we would like to determine .U S(Wn − X) where X is any set of spokes. Problem 4 Develop a technique that results in shorter and more succinct proofs of Theorems 4 and 5. It is likely that such a technique could be applied to a larger family of graphs and advance our understanding of the uniform spectrum in general.

References 1. G. Chartrand, D. Olejniczak, and P. Zhang, Uniformly Connected Graphs. Ars Combin. 154 (2021) 109–124. 2. N. Almohanna, D. Olejniczak, and P. Zhang, Hamiltonian-Connected Graphs with Additional Properties. Congressus Numerantium. 231 (2018) 291–302. 3. G. Chartrand, P. Zhang & T.W. Haynes (Communicated by) (2004) Distance in Graphs - Taking the Long View, AKCE International Journal of Graphs and Combinatorics, 1:1, 1–13, https:// doi.org/10.1080/09728600.2004.12088775

Strongly Regular Multigraphs Leah H. Meissner and John T. Saccoman

Abstract A family of regular graphs which have a direct connection to structures in algebraic combinatorics are strongly regular graphs. These graphs are defined by 4 parameters, n, k, a, and c, where n is the number of nodes, k is the degree of each node, a is the number of common neighbors for every adjacent pair of nodes, and c is the number of common neighbors for every nonadjacent pair of nodes. A multigraph is a graph that has no self-loops, but may have multiple edges and is formally defined by specifying a graph G and assigning a multiplicity to each edge of G. We examine underlying strongly regular multigraphs in order to further clarify their properties, specifically with regard to combinatorial configurations. Keywords Strongly regular graphs · Multigraphs · Algebraic combinatorics

1 Introduction In this chapter, for graph theory notation and terminology, we use [2] and for algebraic combinatorics concepts, we use [3], unless otherwise specified. A strongly regular graph is a regular graph with parameters .(n, k, a, c), such that n is the number of nodes, k is the degree of regularity (.0 < k < v − 1), a is the number of common neighbors for every pair of adjacent nodes, and c is the number of neighbors for every pair of nonadjacent nodes. To illustrate this concept, we consider the strongly regular graph .K3,3 . This simple graph has strongly regular parameters .(6, 3, 0, 3); i.e., there are 6 nodes, all of degree 3, and adjacent nodes have no common neighbors, while nonadjacent nodes have 3 common neighbors. We consider the adjacency matrix .A = aij of a simple graph G such that .A is defined as follows:

L. H. Meissner · J. T. Saccoman (□ ) Department of Mathematics and Computer Science, Seton Hall University, South Orange, NJ, USA e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 S. Heuss et al. (eds.), Combinatorics, Graph Theory and Computing, Springer Proceedings in Mathematics & Statistics 462, https://doi.org/10.1007/978-3-031-62166-6_15

205

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Fig. 1 .K3,3

⎧ aij =

.

1, if nodes vi and vj are adjacent 0, otherwise

In Fig. 1, if we label the top three nodes 1,2,3 and the bottom three nodes as 4,5,6, an adjacency mstrix for .K3,3 would take the form: ⎡

0 ⎢0 ⎢ ⎢ ⎢0 .A(K3,3 ) = ⎢ ⎢1 ⎢ ⎣1 1

0 0 0 1 1 1

0 0 0 1 1 1

1 1 1 0 0 0

1 1 1 0 0 0

⎤ 1 1⎥ ⎥ ⎥ 1⎥ ⎥ 0⎥ ⎥ 0⎦ 0

The eigenvalues for this matrix are .α = {−3, 0, 0, 0, 0, 3}. A well-known algebraic graph theory result for strongly regular graphs is presented in Theorem 1.1. Theorem 1.1 Let G be a connected regular graph of degree .k > 0. Then, the n node graph G is strongly regular if and only if it has exactly 3 distinct adjacency eigenvalues, namely .α1 = k (multiplicity 1), .α2 = θ (multiplicity .mθ ), .α3 = τ (multiplicity .mτ ), where .mθ + mτ + 1 = n [1]. Thus, .K3,3 is a strongly regular graph with three distinct adjacency matrix eigenvalues .−3, 0, and 3. Remark 1.2 Strongly regular graphs have adjacency matrix characteristic polynomial .(α − k)1 (α − θ )mθ (α − τ )mτ . The values for .θ and .τ can be derived from the equation: A2 − (a − c)A − (k − c)I = cJ ,

.

which holds for every strongly regular graph, where .A is the adjacency matrix, .I is the identity matrix, and .J is the matrix of all 1s.

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2 Multigraph Considerations We define a .μ-inflation of a graph G to be one in which we assign a multiplicity, μ, where .μ ∈ N and .μ > 1, to every edge of a simple graph in order to arrive at a particular multigraph variation. The adjacency matrix for a multigraph M is defined by

.

⎧ aij =

.

m, where m is the number of edges between nodes vi and vj 0, otherwise

Suppose we wish to create a 3-inflation of .K3,3 ; i.e., every edge of .K3,3 is now a multiple edge of multiplicity 3. The graph would appear as in Fig. 2, and is denoted (3) .K 3,3 , where the superscipt is the value of .μ. (3)

The multigraph .K3,3 has adjacency matrix ⎡

0 ⎢0 ⎢ ⎢ ⎢0 (3) .A(K 3,3 ) = ⎢ ⎢3 ⎢ ⎣3 3

0 0 0 3 3 3

0 0 0 3 3 3

3 3 3 0 0 0

3 3 3 0 0 0

⎤ 3 3⎥ ⎥ ⎥ 3⎥ ⎥. 0⎥ ⎥ 0⎦ 0

The eigenvalues for this matrix are .α = {−9, 0, 0, 0, 0, 9}. Note that it has three distinct eigenvalues; in fact, strongly regular graphs which are uniformly inflated by a factor of .μ will also possess exactly three distinct adjacency eigenvalues. These eigenvalues are .μk, .μθ , and .μτ , accordingly, which leads to Theorem 2.1. Theorem 2.1 Consider the strongly regular graph G with adjacency eigenvalues k, .θ , and .τ . The multigraph M formed by a uniform .μ-inflation of each edge of G by a factor of .μ will have adjacency matrix characteristic polynomial .P (α ' ) = (α ' − μk)1 (α ' − μθ )mθ (α ' − μτ )mτ . Proof Let M a multigraph that is a uniform .μ-inflation of an underlying strongly regular graph G with multiedge multiplicity .μ. Let .A be the .n × n adjacency matrix for the underlying strongly regular graph. (3)

Fig. 2 .K3,3

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⎡

μa11 − α ' μa12 ⎢ μa21 μa22 − α ' ⎢ .χ (M) = det ⎢ .. .. ⎣ . . μan2 μan1 ⎡

a11 − ⎢ ⎢ a21 n . = μ det ⎢ .. ⎢ ⎣ . an1

α' μ

a12 a22 − .. . an2

α' μ

... ... .. .

μa1n μa2n .. .

⎤ ⎥ ⎥ ⎥ ⎦

. . . μann − α '

... a1n ... a2n .. .. . . . . . ann −

⎤

α' μ

⎥ ⎥ ⎥ ⎥ ⎦

α' , that the matrix is precisely that for the underlying μ strongly regular graph (SRG). Since the underlying simple graph is strongly regular, there are 3 distinct ' ' eigenvalues, namely, .k, θ, τ . Hence, let .α1 = k = αμ1 , .α2 = θ = αμ2 , and

We note that, if we let .α =

'

α3 = τ = αμ3 ⇒ α1 ' = μk, α2 ' = μθ, α3 ' = μτ. So .χ (M) = det|μA − α ' I| = (α ' − μk)1 (α ' − μθ )mθ (α ' − μτ )mτ ; i.e., every uniform .μ-inflation of a strongly █ regular graph G has exactly 3 distinct eigenvalues.

.

3 Designs A particular incidence structure of interest to our inquiry is called a “t-design”. If v, k, t, and .λ are integers such that .0 ≤ t ≤ k ≤ v and .λ ≥ 1, then we have a t-design on v points with blocksize k and index .λ. Accordingly, this is an incidence structure .D = (P, B, I) with .|P| = v, .|B| = k for all .B ∈ B, and for any T set of t points, there are exactly .λ blocks incident with all points in T . Hence, all of the blocks in a given t-design have the same size and every t-subset of the point set is contained in the same number of blocks. We can denote these structures as either .t − (v, k, λ) designs or .Sλ (t, k, v) designs. Furthermore, balanced incomplete block designs, or BIBDs for short, are a kind of t-design with .t = 2. A balanced incomplete block design is a pair .(V , B), where V is a v-set and .B is a collection of b k-subsets of V (blocks) such that each element of V is contained in r blocks and any 2-subset of V is contained in .λ blocks [5]. The name for these incidence structures arises from the fact that .k < v (incomplete) and every pair of distinct points is contained in exactly .λ blocks (balanced) [4].

.

Theorem 3.1 A .(v, k, λ)-BIBD has exactly b=

.

blocks [4].

λ(v 2 − v) vr = k k2 − k

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Theorem 3.2 In a .(v, k, λ)-BIBD, every point occurs in exactly r=

.

λ(v − 1) k−1

blocks [4]. The incidence matrix of a BIBD . = (V , B) with parameters .v, b, r, k, λ is a .v × b matrix .A = (aij ) , in which .aij = 1 when the ith element of V occurs in the j th block of .B, and .aij = 0 otherwise [5]. Theorem 3.3 (Fisher’s Inequality) If a .(v, k, λ)-BIBD exists with .2 ≤ k < v, then b ≥ v [5].

.

To illustrate how to construct a graph from a given design, we can look at of one of the most famous graphs, the Petersen graph, which also happens to be strongly regular. The Petersen graph can be derived from the unique .2 − (6, 3, 2)-block design [5]. That is, the Petersen graph is a .(6, 3, 2)-BIBD with 6 points, 10 blocks, a repetition number of 5, and blocksize 3, where every set of two distinct points is incident with exactly two blocks. One way to arrive at the graphical representation of the Petersen graph is to let the nodes of the graph represent the blocks of the design, where two nodes are adjacent if the corresponding blocks meet in exactly 2 points [5]. If we let .V = {0, 1, 2, 3, 4, 5}, and label the blocks with single letters B = {A = 012, B = 013, C = 024, D = 035, E = 045, F = 125,

.

G = 134, H = 145, I = 234, J = 235}, we arrive at Fig. 3: An alternate approach to construct the Petersen graph is to have all possible 2subsets of the point set .V = {1, 2, 3, 4, 5} correspond to the 10 nodes and draw an edge between two nodes if their labels are disjoint. B = {12, 13, 14, 15, 23, 24, 25, 34, 35, 45},

.

leading to Fig. 4: Fig. 3 Petersen construction from a (6,3,2)-design

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Fig. 4 Alternate Petersen construction

Now that we have outlined these two methods for generating the Petersen graph from a block design, we will look at how block designs give rise to strongly regular graphs from a more general perspective. Definition 3.4 A .t − (v, k, λ) design D is quasi-symmetric with intersection numbers x and y (.x < y) if any two blocks of D intersect in either x or y points [5]. Theorem 3.5 Two distinct blocks A and B of a .2 − (v, k, λ)-design intersect in at least .k − r + λ points [7]. Theorem 3.6 Let D be a quasi-symmetric .2 − (v, k, λ) design with intersection numbers x and y. Then y − x divides .k − x and .r − λ; k(r − 1)(x + y − 1) + xy(1 − b) = k(k − 1)(λ − 1); if .x = 0, then .(r − 1)(y − 1) = (k − 1)(λ ( )− 1); if D has no repeated blocks, then .b ≤ v2 , with equality if and only if D is a 4-design; 5. if .x = 0, then .b ≤ v(v − 1)/k, with equality if and only if D is a 3-design [5]. 1. 2. 3. 4.

. .

Note: any .2 − (v, k, 1) design where .b > v is quasi-symmetric, with corresponding intersection points .x = 0 and .y = 1. Definition 3.7 The block graph .Γ of a quasi-symmetric .2 − (v, k, λ) design D is the graph with node set being the blocks of D, and where blocks .Bi and .Bj are adjacent if and only if .|Bi ∩ Bj | = y [5]. The block graph of a design .D and the block graph of .D, the complement of the design .D, are isomorphic. If every two distinct blocks of a quasi-symmetric BIBD intersect in either .μ1 or .μ2 elements, then the block intersection graph is strongly regular [5]. In turn, this implies the block graph will possess 3 distinct eigenvalues, the smallest of which has been found to be given by the following k−z formula: .−m = − y−x [6]. Based on results from [5], we are able to define the parameters for any strongly regular graph G associated with a given .2 − (v, k, λ)-QSD having intersection

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numbers x and y. These parameters are .(n, k0 , a, c) where .θ and .τ are eigenvalues. Hence, • • • • •

n=b k0 = k(r−1)−x(b−1) y−x .a = k0 + τ θ + τ + θ .c = k0 + τ θ (r−λ−k+x) .θ = y−x . .

• .τ =

x−k y−x

It is worth noting that not every strongly regular graph is a block graph of a quasisymmetric design; i.e. the converse of the previous theorem does not necessarily hold. But, the first construction of the Petersen graph described previously, where two nodes are adjacent if the corresponding blocks meet in exactly two points, will always yield a strongly regular graph if the design is quasi-symmetric. A full parametric classification of quasi-symmetric designs with feasible block graph parameters having smallest eigenvalue .−3 or second eigenvalue 2 can be found in [6]. The authors also include the Mathematica code they used to determine the existence of these designs. An example of such a QSD with second eigenvalue 2 is the (6,2,1)-design, which corresponds to the strongly regular graph with parameters .(15, 8, 4, 4). This graph is the line graph of .K6 (denoted .L(K6 )), or equivalently T(6), which is the triangular graph where .n = 6. This design has intersection points .x = 0, y = 1, meaning that there is an edge drawn when two given blocks meet in one point (.y = 1). Since the number of blocks correspond to the number of nodes, .b = 15 in this case. So, based on these parameters, the blocks are 01, 02, 03, 04, 05, 12, 13, 14, 15, 23, 24, 25, 34, 35, 45. We show Fig. 5, which is a possible labeling as one of the representations for the graph of this structure. Fig. 5 (15,8,4,4)-SRG

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Definition 3.8 A BIBD with .b = v (.r = k) is a symmetric .(v, k, λ)-design in which every two distinct blocks have .λ points in common [5]. A classic example of such a symmetric design construction is the Fano Plane, which is a .(7, 3, 1)-design. If D is a multiple of a symmetric .2 − (v, k, λ)-design, then D is quasi-symmetric with .x = λ and .y = k.

4 Implications for Multigraphs We are not only able to define block graphs in terms of the simple graph setting, but also in the multigraph setting as well. Consider the .(10, 4, 2)-QSD that corresponds to the .(15, 6, 1, 3)-SRG with least eigenvalue .−3. This graph is referred to as the complement of .L(K6 ), or the generalized quadrangle (2,2) (.GQ(2, 2)). The intersection points for this example are .x = 1, y = 2. Hence, for the simple graph, we draw an edge between two nodes if they have 2 points in common. There are 3 non-isomorphic ways to derive the 15 blocks for this design. Here, we use the following realization: 0123, 0145, 0246, 0378, 0579 , 0689, 1278, 1369, 1479, 1568, 2359, 2489, 2567, 3458, 3467. Figure 6 is a representation of one such solution. There are two different approaches that can used in order to generate a particular multigraph variation, depending on the parity of v. If the number of points, v, is even, we can employ a bijection of the form: .zi ↔ zj , where .zi + zj = v − 1. For the case of an odd v, we can use the following: .zi → zj + p, where .1 ≤ p ≤ v − 1. Note: The mapping process for an odd v can be applied to an even case, but the reverse is not true. Thus, it follows that we can use the .(10, 4, 2)-QSD to illustrate both mapping procedures in order to give rise to a strongly regular multigraph. For a mapping in a single direction, this process takes the form: Fig. 6 (15,6,1,3)-SRG

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213 0 .→ 1 1 .→ 2 2 .→ 3 3 .→ 4 4 .→ 5 5 .→ 6 6 .→ 7 7 .→ 8 8 .→ 9 9 .→ 0

0 .→ 2 1 .→ 3 2 .→ 4 3 .→ 5 4 .→ 6 5 .→ 7 6 .→ 8 7 .→ 9 .8 → 0 9 .→ 1

0 .→ 3 1 .→ 4 2 .→ 5 3 .→ 6 4 .→ 7 5 .→ 8 6 .→ 9 7 .→ 0 .8 → 1 9 .→ 2

The blocks resulting from the mapping performed on the values in the first column are 1234, 1256, 1357, 1489, 1680, 1790, 1389, 2470, 2580, 2679, 3460, 3590, 3678, 4569, 4578, which correspond to the multigraph inflation where .μ = 2. So, all of the edges are doubled accordingly because these blocks mirror the structure of the underlying design. To find the blocks associated with the other possible inflations, simply perform the rest of the possible mappings using the other columns. (The table above is not exhaustive.) In turn, the multigraphs for .μ = 3, .μ = 4, etc., can be generated. This will result in .v − 1 associated schemes, as they arise from the .v − 1 possible mappings. Hence, different multigraph variations can be created in this manner by adding the appropriate number of multiple edges, based on the number of mappings performed up to .μ = (v − 1) + 1 = v. This bound indicates that there is a clear maximum number of multiple edges that can occur between any two given nodes in these multigraphs. For the bijection, the 15 blocks of the given QSD become: 9876, 9854, 9753, 9621, 9420, 9310, 8721, 8630, 8520, 8431, 7640, 7510, 7432, 6541, and 6532. This is the case because the mapping .zi ↔ zj where .zi + zj = v − 1 can be attained by the solution .0 ↔ 9, .1 ↔ 8, .2 ↔ 7, etc. These new blocks will again necessarily give rise to a duplicate set of the original edges, as the mapping preserves the underlying block structure. Thus, this process leads to the strongly regular block multigraph, shown in Fig. 7.

5 Future Work The work on strongly regular multigraphs can move forward in several directions. It is unknown if there exist association schemes that yield strongly regular multigraphs other than those that arise from inflating strongly regular simple graphs. In addition, the spectral properties of the Seidel matrix for strongly regular multigraphs also can be explored further.

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Fig. 7 Inflation of the (15,6,1,3)-SRG, .μ = 2

References 1. D. Cvetkovi´c, P. Rowlinson, S. Simi´c An Introduction to the Theory of Graph Spectra London Mathematical Society, Student Texts 75 (2010). 2. G. Chartrand and P. Zhang. A First Course in Graph Theory Dover Publications, Inc. (2012). 3. J.H. van Lint and R.M. Wilson. A Course in Combinatorics Cambridge University Press. (1992). 4. D. Stinson. Combinatorial Designs: Constructions and Analysis Springer. (2004). 5. C.J. Colbourn and J.H. Dinitz. Handbook of Combinatorial Designs Chapman and Hall/CRC. (2007). 6. S.M. Nyayate, R.M. Pawale, and M.S. Shrikhande. Characterization of quasi-symmetric designs with eigenvalues of their block graphs Australian Journal of Combinatorics, vol. 68, no. 1, Dec 2017. 7. A. Neumaier. Quasi-Residual 2-Designs, 1 12 - Designs, and Strongly Regular Multigraphs Geometriae Dedicata, vol. 12, 1982.

Geodesic Leech Graphs Seena Varghese, Aparna Lakshmannan S., and S. Arumugam

Abstract Let .f : E → {1, 2, 3, . . . } be an edge labeling of G. The weight of a path P is the sum of the labels assigned to the edges of P . The edge labeling f is called a geodesic Leech labeling, if the set of weights of the geodesic paths in G is .{1, 2, 3, . . . , tgp (G)}, where .tgp (G), the geodesic path number of G, is the number of geodesic paths in G. A graph which admits a geodesic Leech labeling is called a geodesic Leech graph. In this chapter, we prove the existence of infinite families of geodesic Leech graphs. It is also proved that .C5 is not a geodesic Leech graph. Some open problems in this area are also included. Keywords Leech labeling · Geodesic path number · Geodesic Leech labeling · Geodesic Leech graph AMS Subject Classification: 05C78

1 Introduction By a graph .G = (V , E) we mean a finite undirected graph with neither loops nor multiple edges. The order .|V | and the size .|E| are denoted by n and m respectively. For graph theoretic terminology we refer to Chartrand and Lesniak [2].

S. Varghese Department of Mathematics, Christ College (Autonomous), Irinjalakuda, Kerala, India e-mail: [email protected] Aparna Lakshmannan S. (□) Department of Mathematics, Cochin University of Science and Technology, Cochin, Kerala, India e-mail: [email protected] S. Arumugam National Centre for Advanced Research in Discrete Mathematics, Kalasalingam University, Krishnankoil, Tamil Nadu, India e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 S. Heuss et al. (eds.), Combinatorics, Graph Theory and Computing, Springer Proceedings in Mathematics & Statistics 462, https://doi.org/10.1007/978-3-031-62166-6_16

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Fig. 1 Leech trees

Let .f : E → {1, 2, 3, . . . } be an edge labeling of G. The weight of a path P in G is the sum of the labels of the edges of P and is denoted by .w(P ). Leech [5] introduced the concept of a Leech tree utilizing the unique property of a tree that there exists a unique path between every pair of vertices. Let T be a tree of order n. An edge labeling .f : E → {1, 2, 3, . . . } is called a Leech labeling if the weights of the .n C2 paths in T are exactly 1, 2,. . . ,.n C2 . A tree which admits a Leech labeling is called a Leech tree. Since each edge label is the weight of a path of length one, it follows that f is an injection and 1, 2 are edge labels for all .n ≥ 3. Leech gave examples of five Leech trees (Fig. 1) in [5] itself and these are the only known Leech trees till date. Some variations of Leech trees such as modular Leech trees [3, 4], minimal distinct distance trees [1] and leaf-Leech trees [7] have been investigated by several authors. In [8], we have extended the concept of Leech labeling to the class of all graphs in two different ways each of which is interesting for various reasons. In this chapter we focus on one of the extensions, namely geodesic Leech labeling. The geodesic path number .tgp (G) of a graph G [8] is the total number of geodesic paths in a graph G. Let .f : E → {1, 2, 3, . . . } be an edge labeling of G. If the set of weights of the .tgp (G) geodesic paths in G is .f : E → {1, 2, 3, . . . , tgp (G)}, then f is called a geodesic Leech labeling of G and a graph which admits a geodesic Leech labeling is called a geodesic Leech graph [8]. A graph G for which there exists a unique geodesic path between every pair( of ) vertices is called a geodetic graph [6]. Hence, for geodetic graphs G, .tgp (G) = . n2 . The value of .tgp (G) for various families of graphs is given in [8] and it is observed that cycles of length 3 and 4, .Kn and .Kn − e, where e is any edge, for every n are geodesic Leech graphs. The Geodesic Leech Labeling of .C3 and .C4 is given in Fig. 2. In this chapter, we find geodesic path number of certain graph classes, exhibit infinite classes of geodesic Leech graphs and prove that cycle on 5 vertices is not a geodesic Leech graph.

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Fig. 2 Geodesic Leech labeling of .C3 and .C4

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2 Infinite Classes of Geodesic Leech Graphs A major defect of Leech labeling for trees was that only five Leech trees were identified till date and according to Leech tree conjecture we don’t expect to find more Leech trees. In this section, we establish that the extension of Leech trees to geodesic Leech graphs do not have this limitation by identifying four infinite families of graphs which are geodesic Leech graphs. Lemma 1 Let G be the graph .Kn − {e1 , e2 , . . . , ek }, where .e1 , e2 , . . . , ek are edges on the same vertex. The total number of geodesic Leech paths, .tgp (G) is (incident n) . + (n − k − 2)k. 2 Proof The diameter of G is 2 and hence the geodesic paths will be of length(1)or 2. The number of geodesic paths of length 1, i.e.; the number of edges in G is . n2 − k and the number of geodesic paths of length 2 in G is .(n − k − 1)k. Hence the lemma. Theorem 1 Let G be the graph .Kn − {e1 , e2 , . . . , ek }, where .e1 , e2 , . . . , ek are edges incident on the same vertex. Then G is a geodesic Leech graph for .k ≤ n − 2. Proof Let .V (G) = {u1 , u2 , . . . , un } and let .ei = u1 ui+1 for .i = 1, 2, . . . , k. The geodesic paths in G are the edges and the .u1 ui uj paths of length 2 for .i = k + 2, k + 3, . . . , n and .j = 2, 3, . . . , k + 1. Define an edge labeling f on the edge set of G as follows. Let .f (u1 uk+i+1 ) = i, for .i = 1, 2, . . . , n − k − 1. Now, iteratively assign .f (ui uk+2 ) the smallest number which is not yet obtained as a geodesic path weight, for .i = 2, 3, . . . , k + 1. Repeat the same for each .ui uj for .j = k + 3, k + 4, ..., n. Note that, at each stage, the weight of the geodesic path .u1 ui uj will be greater than the path weights of geodesic paths obtained before assigning the label to .ui uj . After assigning label to .uk+1 un , we get the path weights of .n−k−1+k(n−k−1) edges (geodesic paths of length 1) and .k(n−k−1) geodesic paths of length 2. Therefore, .(n − k − 1) + 2k(n − k − 1) = (n − k − 1)(2k + 1) distinct geodesic path weights are obtained. Hence, .f (uk+1 un ) will be less than or equal to .(n − k − 1)(2k + 1) − 1, so that, the maximum path weight obtained so far is less than or equal to .(n − k − 1)(2k + 1) − 1 + (n − k − 1) = (n(−)k − 1)(2k + 2) − 1. Now, if .(n−k −1)(2k +2)−1 ≤ tgp (Kn −{e1 , e2 , . . . , ek }) = n2 +(n−k −2)k, the remaining path weights can be assigned to the edges not yet labeled in any order to get the geodesic Leech labeling of G. On simplification we get .n2 −n(5+2k)+2k 2 + 4k + 6 ≥ 0, the left hand side of which is a quadratic expression whose discriminant

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is .−4k 2 +4k +1 which is negative for all .k ≥ 2. Therefore, for a fixed k, the value of the quadratic expression is either greater than zero for every n or is less than zero for every n. In particular take .n = 0. Therefore the expression reduces to .2k 2 + 4k + 6 which is positive for every k. Hence, .n2 − n(5 + 2k) + 2k 2 + 4k + 6 ≥ 0 is true for every n and every .k ≥ 2. Therefore, this labeling gives a geodesic Leech labeling for .2 ≤ k ≤ n − 2. (Note that the graph becomes disconnected when .k = n − 1.) The case .k = 1 is proved in [8]. Hence the theorem. Lemma 2 Let G be a graph on n vertices ( ) obtained by attaching k pendent vertices to a vertex of .Kn−k . Then, .tgp (G) = n2 . Proof The graph G is geodetic and hence the lemma follows. Consider the sequence .{an } defined by the recurrence relation .an = an−1 + .n ≥ 2, where .a0 = 0, a1 = 1. Solving this recurrence relation, we an−2 + 1, for √ n √ √ n √ √ )( 1+ 5 ) + ( 5−3 √ )( 1− 5 ) − 1. We use this sequence to define an get, .an = ( 5+3 2 2 2 5 2 5 edge labeling of G in the following theorem. Theorem 2 Let G be the graph obtained by attaching k pendent vertices to a vertex of .Kn−k . Then G is a geodesic Leech graph, if n and k satisfies the relation .ak+1 + √ √ √ k √ k ( ) √ )( 1+ 5 ) + ( 5−3 √ )( 1− 5 ) − 1. (n − k − 1)ak + n − k − 2 ≤ . n2 , where .ak = ( 5+3 2 2 2 5

2 5

Proof Let .V (G) = {u1 , u2 , . . . , uk , v1 , v2 , . . . , vn−k }, where .vi ’s induce .Kn−k and .uj ’s are the pendent vertices incident on .v1 . Define an edge labeling f on edge set of G as follows. Let .f (v1 ui ) = ai , 1 ≤ i ≤ k and .f (v1 vj ) = ak+1 +(ak +1)(j −2), for .j = 2, 3, . . . , n − k. By definition of the sequence .{an } the path weights of all geodesic paths obtained at this stage are all distinct and the maximum path weight attained is .a(k+1 ) + (ak + 1)(n − k − 2) + ak . If this maximum path weight is less than or equal to . n2 , which is the .tgp (G), then we can assign the remaining labels to the remaining edges (which are not part of any geodesic path other than the edge itself) to get the geodesic Leech labeling of G. Hence the theorem. Note The √ relation between n and k in the above theorem can be simplified as .n ≥ (2ak +3)+ 4ak 2 +(4−8k)ak +8ak+1 −(8k+7) . The following result is from [8]. 2 Lemma 3 ([8]) The number of geodesic Leech paths, .tgp (Kn −{e1 , e2 }),.n ≥ 4 ( total ) in .Kn − {e1 , e2 } is . n2 + 2n − 6, where .e1 and .e2 are independent edges. Theorem 3 Let G be the graph obtained by deleting two independent edges .e1 and e2 from .Kn . Then G is a geodesic Leech graph .

.

Proof Let .V (Kn − {e1 , e2 }) = {u, v, w, x, u1 , u2 , . . . , un−4 } and .e1 = uv, e2 = wx. Define .f (uw) = 1, f (vx) = 2, f (ux) = 3 and .f (vw) = 6 (which is the geodesic Leech labeling of .C4 ). Assign the labels from 9 to 2n to .uu1 , uu2 , . . . , uun−4 , .wu1 , wu2 , . . . , wun−4 , respectively, so that all path weights up to 2n are attained. Now, continue the labeling process by iteratively assigning the smaller number which is not yet a path weight to .vu1 , vu2 , . . . , vun−4 , xu1 , xu2 , . . . , xun−4 , respectively. At this stage we get .6(n − 4) + 8 = 6n − 16 path weights

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Table 1 Edge labeling extendable to a geodesic Leech labeling Number of vertices n=5 n=6 n=7 n=8

Edges labels 2, 4, 5, 6, 12, 1, 11, 3 3, 8, 7, 6, 20, 5, 1, 13, 17, 4, 2, 12 1, 2, 3, 21, 6, 8, 10, 18, 19, 16, 11, 13, 14, 17, 12, 15 1, 2, 3, 6, 9, 10, 11, 12, 17, 18, 19, 24, 13, 14, 15, 16, 21, 23, 20, 22

Maximum path weight obtained 14 21 29 38

.tgp (G)

14 21 29 38

of geodesic paths. Hence the label assigned to .xun−4 will be less than or equal to .6n − 17. So, the maximum path weight obtained will be less than or equal to .2n + 6n − 17 = 8n − 17. Therefore, we can assign edge labels which are not yet path weights to the remaining ( ) edges of the form .ui uj , i, j ∈ {1, 2, . . . , n − 4}, provided .8n − 17 ≤ tgp (G) = n2 + 2n − 6. This is true for all .n ≥ 11. For .n = 10, when we actually follow the procedure up to labeling .xu6 with 35, the maximum path weight of a geodesic path obtained is 55 which is less than the .tgp (G) = 59. Similarly, for .n = 9, we label .xu5 with 29 to get maximum path weight as 47 which is less than .tgp (G) = 48. For .5 ≤ n ≤ 8, we cannot obtain geodesic Leech labeling of G using this procedure. Table 1 gives an edge labeling of the edges in the order .uw, vx, vw, ux, uu1 , . . . uun−4 , vu1 , . . . vun−4 , wu1 , . . . wun−4 , xu1 , . . . xun−4 for .5 ≤ n ≤ 8 which can be extended to a geodesic Leech labeling of the respective graph G by assigning the missing path weights to the remaining edges. Hence the theorem. Theorem 4 Let G be the graph obtained by identifying a vertex of two complete graphs .Km and .Kn . Then G is a geodesic Leech graph. Proof Let G be the graph obtained by identifying a vertex of two complete graphs Km and .Kn . Let .V (Km ) = {u1 , u2 , . . . , um } and .V (Kn ) = {v1 , v2 , . . . , vn } where .u1 = v1 . Let f be an edge labeling defined on the edge set of G as follows. Let .f (u1 ui ) = i − 1, for .i = 2, 3, . . . , m and .f (v1 vj ) = (j − 1)m, for .j = 2, 3, . . . , n. This will give all geodesic path (weights)from 1 to .mn − 1. The remaining edges may be assigned values from mn to . m+n−1 , which is the .tgp (G), since G is geodetic. 2 .

Theorem 5 Let G be the graph on n vertices obtained by identifying one vertex each of k complete graphs .Km1 , Km2 , . . . , Kmk , where .m1 ≥ m(2 )≥ · · · ≥ mk . Then G is a geodesic Leech graph if .m1 m2 ...mn−2 (mn−1 mn − 1) ≤ n2 . Proof Let G be the graph obtained by identifying one vertex each of k complete graphs .Km1 , Km2 , . . . , Kmk . Let .V (Kmi ) = {ui1 , ui2 , . . . , uimi }, for .i = 1, 2, . . (. ,)k where .ui1 = u, for every i. Note that G is a geodetic graph and hence .tgp (G) = n2 . Define a labeling f on the edge set as .f (uu1i ) = i − 1, for .i = 2, 3, . . . , m1 ; .f (uu2i ) = (i − 1)m1 , for .i = 2, 3, . . . m2 , .. . . , .f (uuni ) = m1 m2 ...mk−1 (i − 1), for .i = 2, 3, . . . , mk . Now, the edges corresponding to all geodesic paths of length

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two are labeled, the geodesic path weights obtained till now are all different and the maximum path weight obtained is .m1 m2 ...mn−1 (mn − 1) + m1 m2 ...mn−2 (mn−1 − 1) = m1 m2 ...mn−2 (mn−1 mn − 1). If this maximum path weight is less than or equal to .tgp (G), the labeling can be extended to a geodesic Leech labeling by assigning the remaining path weights to the remaining edges. Hence the theorem.

3 Concluding Remarks and Open Problems The graphs which do not admit a geodesic Leech labeling are termed as nongeodesic Leech graphs. We have seen that there are many infinite families of graphs which are geodesic Leech graphs. The complement class is also expected to be infinite. Though, .C4 = K2,2 is a geodesic Leech graph, we believe that .Cn and .Km,n are non-geodesic Leech graphs in all other cases. Since odd cycles ( ) are geodetic, geodesic path number is . n2 itself and for even cycles it is an easy 2

and when .m = n, we observation that .tgp (Cn ) = n2 . Also, .tgp (Km,n ) = mn(m+n) 2 get .tgp (Kn,n ) = n3 . Here we include a proof for .C5 is a non-geodesic Leech graph. Theorem 6 The cycle .C5 is not a geodesic Leech graph. Proof We have .tgp (C5 ) = 10. In .C5 the geodesic paths are precisely edges and paths of length 2. If possible assume that .C5 admits a geodesic Leech labeling. We know that 1 and 2 must be edge labels in any geodesic Leech labeling. If 1 and 2 are assigned to two adjacent edges then the remaining edge labels must be at least 4, 5 and 6. The edges labeled 5 and 6 (or any two higher values) cannot be adjacent, since .tgp (C5 ) = 10. Since all path weights up to 6 are to be attained and repetition of path weights is not permitted, the labels must be cyclically 1, 2, 5, 4, 6 or 1, 2, 7, 4, 5 both of which are not geodesic Leech labeling. Therefore, 1 and 2 must be assigned to non-adjacent edges. If 3 is assigned to an edge adjacent to both the edges labeled 1 and 2, then the remaining two adjacent edges must have label 6 and 7 or higher, which is a contradiction. If 3 is adjacent to the edge labeled 1 alone, then the labels must be cyclically 1, 3, 8, 2, 5 which is not a geodesic Leech labeling. If 3 is assigned to an edge adjacent to the edge labeled 2 alone, then the path weight 4 cannot be attained without repeating the path weight 2+3 = 5. Hence, .C5 is not a geodesic Leech graph. It is easy to prove .C6 and .C7 are non-geodesic Leech graphs in a similar fashion. But, the general problem seems interesting. Problem 1 Prove that .Cn is a non-geodesic Leech graph for .n ≥ 5. Problem 2 Prove that .Km,n is a non-geodesic Leech graph except for .K2 , P3 , C4 and .K1,3 .

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A much more general problem is to characterize geodesic Leech graphs. Also, if we look into an edge labeling of .C5 in which edges are labeled cyclically 1, 3, 7, 2 and 5, you can see that path weight 7 is repeated and 8 is missing. Such labelings in which exactly one path weight is missing and one path weight is repeated were termed as almost Leech trees in [7]. In a similar fashion one can define almost geodesic Leech graphs and .C5 is an example of such a graph. Acknowledgments The second author thank Cochin University of Science and Technology for granting project under Seed Money for New Research Initiatives (order No.CUSAT/PL(UGC).A1/1112/2021) dated 09.03.2021.

References 1. B. Calhoun, K. Ferland, L. Lister, and J. Polhill, Minimal distinct distance trees, Journal of Combinatorial Mathematics and Combinatorial Computing, 61, 33–57 (2007). 2. G. Chartrand and L.Lesniak, Graphs and digraphs, CRC (2005). https://doi.org/10.1201/b19731 3. D. Leach, Modular Leech trees of order atmost 8, J. Combin. (2014), Article ID 218086. https:// doi.org/10.1155/2014/218086 4. D. Leach and M. Walsh, Generalized Leech trees, J. Combin. Math. Combin. Comput. 78, 15–22 (2011). 5. J. Leech, Another tree labeling problem, Amer. Math. Monthly, 82, 923–925 (1975). https://doi. org/10.1080/00029890.1975.11993981 6. O. Ore, Theory of Graphs, Colloquium Publications (1962). 7. M. Ozen, H. Wang, D. Yalman, Note on Leech-type questions of tree, Article in Integers 16 (2016). 8. Seena Varghese, Aparna Lakshmanan S, S. Arumugam, Two extensions of Leech labeling to the class of all graphs, AKCE Int. J. Graphs Comb., 19(2), 159–165 (2022). https://doi.org/10.1080/ 09728600.2022.2084354

Characterizing s-Strongly Chordal Graphs Using 2-Paths and k-Chords Terry A. McKee

Abstract A well-known 1961 characterization of chordal graphs by G. A. Dirac in terms of simplicial vertices, was extended by Chvátal, Rusu, and Sritharan (Discrete Math., 2002) to a natural sequence of “weakly chordal graphs.” Somewhat similarly, we will start from the same place and characterize a natural sequence of “chordal, strongly chordal, . . . , s-strongly chordal, . . . ” graphs, except now in terms of 2-path graphs (meaning graphs that are either .K3 or a 2-tree graph with exactly two degree2 vertices—equivalently, “paths of triangles” that are the 2-connected outerplanar strongly chordal graphs). Keywords Chordal graph · Strongly chordal graph · k-Chord · 2-Path MSC Classification 05C75 (05C38)

1 Introduction Define the length of a path or cycle to be the number of edges in it, so length-k paths have .k + 1 distinct vertices, and length-0 paths are vertices. Define a k-cycle to have length k and a .≥ k-cycle to have length at least k. Define the interior subpath of a length-k path .(v0 , v1 , . . . , vk−1 , vk ) with .k ≥ 2 to be the length-.(k − 2) subpath .(v1 , . . . , vk−1 ). Define a chord of a cycle C to be an edge xy that has .x, y ∈ V (C) with .xy /∈ E(C), and define it to be an i-chord if x and y are a |distance i apart in | the cycle C; thus, the i-chords of a k-cycle always have .2 ≤ i ≤ k2 . A graph is chordal if every .≥ 4-cycle has at least one chord, which in fact is equivalent to every .≥ 4-cycle having at least one 2-chord [8]. Define a graph to be k-chordal if every .≥ k-cycle has a chord, making the 4-chordal graphs precisely the

T. A. McKee (□) Wright State University, Dayton, OH, USA e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 S. Heuss et al. (eds.), Combinatorics, Graph Theory and Computing, Springer Proceedings in Mathematics & Statistics 462, https://doi.org/10.1007/978-3-031-62166-6_17

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chordal graphs. (This definition of k-chordal agrees with [2], but is sometimes called “.(k − 1)-chordal,” for instance in [3, 7].) A vertex v is a simplicial vertex of G if every two neighbors of v are adjacent in G. One of the earliest in the galaxy of characterizations of a graph being chordal [2, 13] is due to Dirac in 1961 [5]—that every induced subgraph has a simplicial vertex. In the 2002 paper [3], Chvátal, Rusu, and Sritharan start from Dirac’s characterization and restate it as G being chordal (.= 4-chordal) if and only if each nonempty induced subgraph H of G contains a length-0 path .π0 (a vertex) that is not the interior subpath of any induced length-2 path of H (in other words, .π0 is a simplicial vertex of H ). Reference [3] then generalizes this to characterize when a graph is kchordal (also see [7]). Proposition 1 will restate the generalized characterization from [3], with its .k = 1 case corresponding to [5], or see [2, 13]. (All the .k ≥ 1 cases are proved in [3], which also contains the history of the .k = 1 case.) Proposition 1 (From [3]) For each .k ≥ 1, a graph is .(k + 3)-chordal if and only if each nonempty induced subgraph H has induced paths of lengths .0, . . . , k − 1 that are not the interior subpaths of induced paths of H that have, respectively, lengths .2, . . . , k + 1. Sections 2 and 3 will go in the opposite direction, away from the weakening of chordal graphs in Proposition 1, toward the strengthening the widely studied “strongly chordal graphs.”

2 s-Strongly Chordal Graphs and 2-Paths Martin Farber defined the strongly chordal graphs in 1983 [6], with most of their characterizations appearing in [1, 2, 6, 13]. For our purposes, it is convenient to define a graph to be strongly chordal if it is chordal and every .≥ 6-cycle has at least one 3-chord [8]. Another characterization, from [6], is that a graph is strongly chordal if and only if it is chordal and no induced subgraph is an n-sun graph, denoted .Sn , formed from the complete graph .Kn with .n ≥ 3, augmented by identifying one edge from each of n additional triangles with an existing edge of a particular spanning n-cycle of the original .Kn . The graphs .S3 and .S4 are shown in Fig. 1. (Note that the Hamiltonian 2n-cycle of an n-sun .Sn never has any i-chords with i odd.) Fig. 1 The 3-sun .S3 (on the left) and 4-sun .S4 graphs

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As in [10], for every .s ≥ 0, define a graph to be s-strongly chordal if, for every .i ∈ {2, . . . , s + 2}, each .≥ 2i-cycle has an i-chord—equivalently, every cycle long enough to have such an i-chord actually does have an i-chord. Thus, the chordal graphs are the 0-strongly chordal graphs, and the strongly chordal graphs are the 1-strongly chordal graphs. Two of the four minimum-order strongly chordal graphs that are not 2-strongly chordal are shown in Fig. 2 (the missing two are the supergraphs of the left graph that are subgraphs of the right graph). Note that neither has any 4-chords. See [10, 12] for more about s-strongly chordal graphs. Theorem 2 will give a new, general characterization of s-strongly chordal graphs, but it first requires the following definition of the “2-tree” and “2-path” graphs. Define a graph G to be a 2-tree (originally called a “dimension-2 tree”) if it can be constructed as follows, as in [11, 14, 15]. First of all, .K3 is a 1-triangle 2-tree. Then, whenever G is a k-triangle 2-tree of order n with .xy ∈ E(G), a new vertex z can be adjoined adjacent to both x and y to form a new .(k + 1)-triangle 2-tree of order .n + 1, and this can be repeated recursively to form larger 2-trees. (Although some papers start from .K2 instead, starting from .K3 is more convenient for our purpose—so a subgraph of order at least 3 of a 2-tree is 2-connected if and only if it is nonseparable.) Finally, define a graph .⨅ to be a 2-path if either .⨅ ∼ = K3 (which can be called its own unique end triangle) or .⨅ has exactly two simplicial, degree-2 vertices (which are in its two end triangles). Figure 3 shows four examples of 2-paths. The 3-sun in Fig. 1 and the graph .K1,1,3 (three triangles sharing one common edge) are the smallest examples of 2-trees that are not 2-paths. A 2-path can be thought of as a “path of triangles.” Every 2-path .⨅ is strongly chordal and outerplanar [9], and when .⨅ has simplicial vertices u and v, its “exterior face” is bordered by two interiorly-disjoint v-to-w paths that form its spanning cycle .C⨅ (with .⨅ having one more triangle than .C⨅ has chords). If .C⨅ ∼ / K3 , then .E(C⨅ ) consists of the two edges of each of the = two end triangles of .⨅ together with one edge from each of the other triangles of .⨅. For instance, the rightmost 2-path .⨅ in Fig. 3 has seven triangles, and .C⨅ is the 9-cycle that has six i-chords where, from left to right, .i = 2, 3, 4, 4, 3, 2. Fig. 2 Two smallest strongly chordal, but not 2-strongly chordal, graphs

Fig. 3 An assortment of 2-path graphs

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As will be used in the proofs in §3, the recursive construction of 2-trees allows the construction of 2-paths to begin with either end triangle of a 2-path .⨅ /= K3 , with one of the two neighbors of a simplicial vertex v of .⨅ always becoming simplicial in .⨅ − v. Moreover, if .⨅ is a 2-path with .n ≥ 3 vertices, then .⨅ has .n − 2 triangles and .C⨅ has .n − 3 chords, with i-chords whenever .2 ≤ i ≤ ⎣ n2 ⎦. Finally, note that every 2-path is s-strongly chordal for all .s ≥ 0.

3 Characterizing s-Strongly Chordal Graphs From now on, each 2-path .⨅ will be called a 2-path subgraph to emphasize that .⨅ is merely a 2-path subgraph that is contained in some given graph—but .⨅ does not have to be an induced subgraph of that graph. The .k = 1 case of Proposition 1 can be re-paraphrased as follows: A graph is chordal if and only if each .≥ 4-cycle C has a length-2 subpath .π2 of C that is also a subpath of a triangle .⨅. This formulation corresponds to the .s = 0 instance of the following characterization of s-strongly graphs, for all .s ≥ 0, in terms of 2-path subgraphs. Theorem 2 For each .s ≥ 0, a graph is s-strongly chordal if and only if, for each k ∈ {0, 1, . . . , s}, each .≥ (2k + 4)-cycle has a length-.(k + 2) subpath .πk+2 that is a subpath of the spanning cycle of a .(k + 1)-triangle 2-path subgraph .⨅ such that the endpoints of .πk+2 are simplicial vertices of .⨅.

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Proof First, suppose G is s-strongly chordal, .0 ≤ k ≤ s, and C is a .≥ (2k + 4)-cycle that has a length-.(k + 2) subpath .πk+2 [toward showing that .πk+2 is a subpath of the spanning cycle of a .(k + 1)-triangle 2-path subgraph .⨅ such that the endpoints of .πk+2 are simplicial vertices of .⨅]. Argue by induction on .s ≥ 0. In the .s = 0 basis step, the interior subpath of .π2 is a vertex, and by Proposition 1, there is a 1-triangle 2-path subgraph .⨅ in G, as described. For the induction step, assume .s ≥ 1 (so by the inductive hypothesis, G is k-strongly chordal whenever .0 ≤ k ≤ s − 1) with .k = s, and let .πk+2 = < = (v0 , v1 , . . . , vk+1 ) (v0 , v1 , . . . , vk+1 , vk+2 ). By the inductive hypothesis, .πk+1 > and .πk+1 = (v1 , .. . . , vk+1 , vk+2 ) are length-.(k + 1) subpaths of, respectively, spanning cycles .C⨅< and .C⨅> of k-triangle (and so, of s-triangle) 2-path subgraphs < > < > .⨅ and .⨅ in G. Because of the overlapping vertex sets of .π k+1 and .πk+1 , the < > subgraphs .⨅ and .⨅ in G that have simplicial vertices, respectively, .v0 and .vk+2 where—as mentioned in the final paragraph of §2—one of the two neighbors of, respectively, .v0 and .vk+2 is simplicial in the subgraphs of G induced by .V (⨅ − vk+2 ), and the remaining .k − 1 triangles are the same in both of the smaller 2-path subgraphs .⨅< and .⨅>. Therefore, .(v0 , . . . , vk+2 ) is a subpath of the .(1 + [k − 1] + 1) = (k + 1)-triangle 2-path subgraph .⨅ in G that contains .πk+2 such that the endpoints of .πk+2 are simplicial vertices of .⨅. Conversely, suppose .s ≥ 0 and .k ∈ {0, 1, . . . , s}, and suppose each .≥ (2k + 4)cycle of G has a length-.(k + 2) subpath .πk+2 as described in the statement of the

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theorem [arguing by induction on .s ≥ 0 that each .≥ (2k + 4)-cycle has a .(k + 2)chord]. The .s = 0 basis step follows by Proposition 1. For the induction step, assume .s ≥ 1 and .0 ≤ k ≤ s − 1, and let C be any ≥ . (2s + 4)-cycle of G. Since .|E(C)| ≥ 2s + 4 > 2[s − 1] + 4 = 2s + 2, the .k = s − 1 inductive hypothesis implies C has a length-.(s + 1) subpath .πs+1 that is a subpath of the spanning cycle .C⨅ of an s-triangle 2-path subgraph .⨅ such that the endpoints of .πs+1 are simplicial vertices of .⨅. Thus, .|E(C⨅ )| = 2s + 4 and—as mentioned in the final paragraph of §2—cycle C has an i-chord whenever .2 ≤ i ≤ ⎣ 2s+4 2 ⎦, and so C has an .(s + 2)-chord. Therefore, each .≥ (2k + 4)-cycle has a .(k + 2)-chord, and so G is s-strongly chordal. ⨆ ⨅ Proposition 3 will translate Corollary 3.2 of [12] into our current terminology. This is another characterization of being s-strongly chordal in terms of 2-path subgraphs, except that [12] is based on an innovative characterization of strongly chordal by Dahlhaus, Manuel, and Miller from 1998 [4]—as opposed to [3] having been based on [5]. Thus, Theorem 2 and Proposition 3 come from somewhat different directions and from the original statements of the .s = 1 and .s = 0 cases in, respectively, [5] and [4]. Proposition 3 ([12]) For each .s ≥ 0, a graph is s-strongly chordal if and only if, for each .k ∈ {0, 1,. . . ,s}, each .≥ (2k + 4)-cycle is the spanning cycle of a .(k + 3)triangle 2-path subgraph .⨅ such that the triangles of .⨅ consist of one or two chords of .C⨅ and, respectively, two or one edges in .E(C⨅ ). After all the original motivation from [5] and [4]—and so from, respectively, [3] and [12]—and after all the specific details disappear, the final, general formulations of Theorem 2 and Proposition 3 remain. Each might be awkwardly viewed as a corollary of the other, but at the cost of obscuring their distinct historical motivations. Their distinctness can be viewed in terms of the former’s focus on the .𝓁 − 1 edges of a subpath of a spanning .2𝓁-cycle of a 2-path subgraph, with the latter’s focus on the .2𝓁 edges of the cycle. (The differing number of triangles in .⨅ in the resulting statements reflect that both length-.n paths and length-.n cycles are traditionally defined in terms of having n edges.) Their distinct historical motivations vanish in the following characterization of s-strongly chordal graphs, which is simultaneously—with essentially the same proof—a corollary of either Theorem 2 or Proposition 3. Corollary 4 For each .s ≥ 0, a graph is s-strongly chordal if and only if, for each k ∈ {0, 1, . . . , s}, each .≥ (2k + 4)-cycle is the spanning cycle of a 2-path subgraph.

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Proof The “only if direction” follows immediately from the only if direction of Theorem 2 or of Proposition 3. For the “if direction,” suppose .s ≥ 0, and G is k-strongly chordal for all .k ∈ {0, 1, . . . , s − 1}, and suppose that each .≥ (2k + 4)-cycle is the spanning cycle of a 2-path subgraph .⨅ where EITHER (to use Theorem 2) the endpoints of the length.(k + 2) subpath of .C⨅ are simplicial vertices of .⨅ OR (to use Proposition 3) the

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triangles of .⨅ consist of one or two chords of .C⨅ and, respectively, two or one edges in .E(C⨅ ). Assume that G is not s-strongly-chordal [arguing by contradiction]. Thus, some ≥ . 2(k + 2)-cycle C of G with .k + 2 ≥ s has no .(k + 2)-chord, and yet has a 2-path subgraph .⨅ with .C = C⨅ as described in the preceding paragraph. Therefore— as mentioned in the final paragraph of §2—there are .2k + 2 edges in .E(C) and .(2k + 4) − 3 = 2k + 1 chords of C in .E(⨅), including at least one i-chord whenever 2k+4 .2 ≤ i ≤ ⎣ 2 ⎦ = k + 2. But now, this .(k + 2)-chord would contradict C having no such chord. ⨆ ⨅

References 1. A. Brandstädt and M. C. Golumbic, Dually and strongly chordal graphs, in Topics in Algorithmic Graph Theory (L. W. Beineke, M. C. Golumbic, and R. J. Wilson, editors) Cambridge University Press (2021) 152–167. 2. A. Brandstädt, V. B. Le, and J. P. Spinrad, Graph Classes: A Survey, Society for Industrial and Applied Mathematics, Philadelphia, 1999. 3. V. Chvátal, I. Rusu, and R. Sritharan, Dirac-type characterizations of graphs without long chordless cycles, Discrete Math. 256 (2002) 445–448. 4. E. Dahlhaus, P. Manuel, and M. Miller, A characterization of strongly chordal graphs, Discrete Math. 187 (1998) 269–271. 5. G. A. Dirac, On rigid circuit graphs, Abh. Math. Sem. Univ. Hamburg 25 (1961) 71–76. 6. M. Farber, Characterizations of strongly chordal graphs, Discrete Math. 43 (1983) 173–189. 7. R. Krithika, R. Mathew, N. S. Narayanaswamy, and N. Sadagopan, A Dirac-type characterization of k-chordal graphs, Discrete Math. 313 (2013), 2865–2867. 8. T. A. McKee, The i-chords of cycles and paths, Discuss. Math. Graph Theory 32 (2012) 607– 615. 9. T. A. McKee, Planarity of strongly chordal graphs, Bull. Inst. Combin. Appl. 8 (2016) 85–91. 10. T. A. McKee, Strengthening strongly chordal graphs, Discrete Math. Algorithms Appl. 8 (2016) #1650002, 8 pp. 11. T. A. McKee Characterizing 2-trees relative to chordal and series-parallel graphs, Theory Appl. Graphs 8 (2021), issue 1, article 4 (7pp). 12. T. A. McKee, Interplay between chords and duality in strongly chordal graph theory, submitted. 13. T. A. McKee and F. R. McMorris, Topics in Intersection Graph Theory, Society for Industrial and Applied Mathematics, 1999. 14. H. P. Patil, On the structure of k-trees, Combin. Inform. System Sci. 11 (1986) 57–64. 15. R. E. Pippert and L. W. Beineke, Characterizations of 2-dimensional trees, [in The Many Facets of Graph Theory], Lecture Notes in Mathematics, Springer, 110 (1969) 263–270.

The Spectrum Problem for the 4-Uniform 4-Colorable 3-Cycles with Maximum Degree 2 Ryan C. Bunge, Saad I. El-Zanati, Julie N. Kirkpatrick, Shania M. Sanderson, Michael J. Severino, and William F. Turner

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Abstract The complete t-uniform hypergraph of order v, denoted .Kv , has a set V with v elements as its vertex set and the set of all t-element subsets of V as its edge set. For the purposes of this work, we define a 4-uniform 3-cycle of maximum degree 2 to be any 4-uniform hypergraph of maximum degree 2 that can be obtained (2) by adding two vertices to each of the three edges in .K3 . There are five such 4uniform hypergraphs up to isomorphism. Two of them have chromatic number 4. We give necessary and sufficient conditions for the existence of a decomposition of the complete 4-uniform hypergraph of order v into these 4-colorable 3-cycles.

1 Introduction A hypergraph H consists of a finite set V of vertices and a finite set E of nonempty subsets of V called hyperedges or simply edges. For a given hypergraph H , we use .V (H ) and .E(H ) to denote the vertex set and the edge set of H , respectively. We call .|V (H )| and .|E(H )| the order and size of H , respectively. The degree of a vertex .u ∈ V (H ) is the number of edges in .E(H ) that contain u. If for each .e ∈ E(H ) we have .|e| = t, then H is said to be t-uniform. Thus t-uniform hypergraphs are generalizations of the concept of a graph (where .t = 2). The hypergraph with vertex set V and with edge set the set of all t-element subsets of V is called the (t) (t) complete t-uniform hypergraph on V and is denoted by .KV . If .v = |V |, then .Kv is called the complete t-uniform hypergraph of order v and is used to denote any

R. C. Bunge · S. I. El-Zanati (□) · M. J. Severino · W. F. Turner Illinois State University, Normal, IL, USA e-mail: [email protected] J. N. Kirkpatrick Cedar Falls High School, Cedar Falls, IA, USA S. M. Sanderson Old Dominion University, Norfolk, VA, USA © The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 S. Heuss et al. (eds.), Combinatorics, Graph Theory and Computing, Springer Proceedings in Mathematics & Statistics 462, https://doi.org/10.1007/978-3-031-62166-6_18

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hypergraph isomorphic to .KV . When .t = 2, we may use .Kv in place of .Kv . If .H ' is a subhypergraph of H , then .H \ H ' denotes the hypergraph obtained from H by deleting the edges of .H '. We may refer to .H \ H ' as the hypergraph H with a hole ' ' .H . The vertices in .H may be referred to as the vertices in the hole. A commonly studied problem in combinatorics concerns decompositions of graphs into edge-disjoint subgraphs. A decomposition of a graph K is a set }.Δ = { {G1 , G2 , . . . , Gs } of subgraphs of K such that . E(G1 ), E(G2 ), . . . , E(Gs ) is a partition of .E(K). If each element of .Δ is isomorphic to a fixed graph G, then .Δ is called a G-decomposition of K. A G-decomposition of .Kv is also known as a Gdesign of order v. A .Kk -design of order v is usually known as a 2-.(v, k, 1) design or as a balanced incomplete block design of index 1 or as a .(v, k, 1)-BIBD. The problem of determining all v for which there exists a G-design of order v is of special interest (see [1] for a survey). The notion of decompositions of graphs naturally extends to hypergraphs. A decomposition of a hypergraph K is a set .Δ = {H1 , H2 , . . . , Hs } of subhypergraphs of K such that .{E(H1 ), E(H2 ), . . . , E(Hs )} is a partition of .E(K). Any element of .Δ isomorphic to a fixed hypergraph H is called an H -block. If all elements of .Δ are H -blocks, then .Δ is called an H -decomposition of K, and we may also say H (t) decomposes K. An H -decomposition of .Kv is called an H -design of order v. The problem of determining all v for which there exists an H -design of order v is called the spectrum problem for H -designs. (t) A .Kk -design of order v is a generalization of 2-.(v, k, 1) designs and is known as a t-.(v, k, 1) design or simply as a t-design. A summary of results on t-designs appears in [21]. A t-.(v, k, 1) design is also known as a Steiner system and is denoted by .S(t, v, k) (see [14] for a summary of results on Steiner systems). Keevash [20] has shown that for all t and k the obvious necessary conditions for the existence of an .S(t, k, v)-design are sufficient for sufficiently large values of v. Similar results were obtained by Glock, Kühn, Lo, and Osthus [15, 16] and extended to include the corresponding asymptotic results for H -designs of order v for all uniform hypergraphs H . These results for t-uniform hypergraphs mirror the celebrated results of Wilson [26] for graphs. Although these asymptotic results assure the existence of H -designs for sufficiently large values of v for any uniform hypergraph H , the spectrum problem has been settled for very few t-uniform hypergraphs with .t > 2. In the study of graph decompositions, a fair amount of the focus has been on G-decompositions of .Kv where G is a graph with a relatively small number of edges (see [1] and [7] for known results). Some authors have investigated the corresponding problem for 3-uniform hypergraphs. For example, in [5], the spectrum problem is settled for all 3-uniform hypergraphs on 4 or fewer vertices. More recently, the spectrum problem was settled in [6] for all 3-uniform hypergraphs with at most 6 vertices and at most 3 edges. In [6], they also settle the spectrum problem for the 3-uniform hypergraph of order 6 whose edges form the lines of the Pasch configuration. Other authors have also considered H -designs where H is a 3uniform hypergraph whose edge set is defined by the faces of a regular polyhedron. (t)

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Let T , O, and I denote the tetrahedron, octahedron, and icosahedron hypergraphs, respectively. The hypergraph T is the same as .K4(3), and its spectrum was settled in 1960 by Hanani [17]. In another paper [18], Hanani settled the spectrum problem for O-designs and gave necessary conditions for the existence of I -designs. Using the approach in [6], the spectrum problem has recently been settled for several individual 3-uniform hypergraphs H . These include when H is a loose mcycle for .3 ≤ m ≤ 5 (see [8, 11, 13]) and when H is a tight 6-cycle [2] or a tight 9-cycle [10]. There are also several articles on decompositions of complete t-uniform hypergraphs (see [4] and [24]) and of t-uniform t-partite hypergraphs (see [22] and [25]) into variations on the concept of a Hamilton cycle. Moreover, there are results on decompositions of 3-uniform hypergraphs into structures known as Berge cycles with a given number of edges (see for example [19] and [23]). We note however that the Berge cycles in these decompositions are not required to be isomorphic. Perhaps the best known result on decompositions of complete t-uniform hyper(t) graphs is a result by Baranyai [3] on the existence of 1-factorizations of .Kmt for all positive integers m. In this chapter we are interested in the spectrum problem for H -designs where H is a 4-colorable 4-uniform 3-cycle of maximum degree 2. For the purposes of this work, we define a 4-uniform 3-cycle of maximum degree 2 to be any 4-uniform hypergraph of maximum degree 2 that can be obtained by adding two vertices to (2) each of the 3 edges in .K3 . There are five such 4-uniform hypergraphs as shown in Fig. 1. Two of these hypergraphs, .H4 and .H5 , have chromatic number 4 and the remaining three have chromatic number at least 5. The spectrum problem for .H5 was settled in [9] where it was shown that there exists an .H5 -decomposition of .Kv(4) if and only if .v ≡ 0, 1, 2, 3, or .6 (mod 9) and .v ≥ 9. Here we show that the same conditions are necessary and sufficient for the existence of .H4 -decompositions of (4) .Kv . Henceforth, we will let .H [v1 , v2{, v3 , v4 , v5 , v6 , v7 , v8 ] denote } the 4-uniform hypergraph .H4 with vertex set . v1 , v2 , v3 , v4 , v5 , v6 , v7 , v8 and edge set { } . {v1 , v2 , v3 , v4 }, {v4 , v5 , v6 , v7 }, {v1 , v2 , v7 , v8 } as illustrated in Fig. 1.

1.1 Additional Notation and Terminology Let .Zn denote the group of integers modulo n. If a and b are integers, we define [a, b] to be .{r ∈ Z : a ≤ r ≤ b}. For any edge-disjoint t-uniform hypergraphs H and .H ' , we use .H ∪ H ' to indicate the hypergraph with vertex set .V (H ) ∪ V (H ' ) and edge set .E(H ) ∪ E(H ' ). Similarly, if H is a hypergraph and r is a nonnegative integer, then an edge-disjoint union of r copies of H is denoted with rH .

.

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H1

H2

v1 v2 H3 v8 v7

v3 v6

v 5 v4

H4

H5

Fig. 1 The five hypergraphs that are 4-uniform 3-cycles with maximum degree 2

We next define some notation for certain types of multipartite-like 4-uniform hypergraphs. Let .A, B, C, D be pairwise disjoint sets. The hypergraph with vertex set .A ∪ B ∪ C ∪ D and edge set consisting of all 4-element sets having exactly one (4) vertex in each of .A, B, C, D is denoted by .KA,B,C,D . The hypergraph with vertex set .A ∪ B and edge set consisting of all 4-element sets having at least one vertex in each of A and B is denoted by .L(4) A,B . The hypergraph with vertex set .A ∪ B ∪ C and edge set consisting of all 4-element sets having at least one vertex in each of .A, B, (4) C is denoted by .LA,B,C . If .|A| = a, .|B| = b, .|C| = c, and .|D| = d, we may use (4)

(4)

(4)

Ka,b,c,d to denote any hypergraph that is isomorphic to .KA,B,C,D , .La,b to denote

.

(4) any hypergraph that is isomorphic to .L(4) A,B , and .La,b,c to denote any hypergraph (4) (4) (4) that is isomorphic to .LA,B,C . We use .K ∪L to denote any hypergraph a,b,c,d b,c,d (4) (4) (4) (4) isomorphic to .KA,B,C,D ∪ LB,C,D . Similarly, we use .La,b,c ∪ Lb,c to denote any (4) (4) hypergraph isomorphic to .LA,B,C ∪ LB,C . It is simple to observe that if A, B, C, D, and .D ' are pairwise-disjoint, then (4) (4) (4) KA,B,C,D∪D ' = KA,B,C,D ∪ KA,B,C,D ' . Thus we have the following lemma.

.

(4) Lemma 1 If a, b, c, d, w, x, y, and z are positive integers, then .Kwa,xb,yc,zd can (4)

be decomposed into wxyz copies of .Ka,b,c,d .

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Similarly, we observe that if A, B, C, and .C ' are pairwise-disjoint, then (4) (4) (4) = LA,B,C ∪ LA,B,C ' ∪ KA,B,C,C ' . Thus we have the following lemma.

(4) .L A,B,C∪C '

(4)

Lemma 2 If a, b, c, x, y, and z are positive integers, then .Lxa,yb,zc can be ( ) ( ) (4) (4) decomposed into xyz copies of .La,b,c , . x2 yz copies of .Ka,a,b,c , .x y2 z copies of (z) (4) (4) .K a,b,b,c , and .xy 2 copies of .Ka,b,c,c .

2 Some Small Examples Next, we give some examples of .H4 -decompositions that are used in proving our main result. For the most part, these decompositions are either cyclic or r-pyramidal as defined in [12]. They were found either by hand or by computer searches. ⎞ ⎛ } { (4) = Z7 ∪ ∞1 , ∞2 and let Example 2 Let .V K9 .B

{ = H [0, 1, 2, ∞1 , 5, 4, ∞2 , 3], H [0, 5, 4, 2, 6, 3, ∞2 , 1], H [2, 3, 1, 0, 4, ∞1 , 5, 6],

} H [1, 2, 4, 0, ∞1 , 5, ∞2 , 3], H [1, 2, 5, 0, 3, 4, ∞1 , 6], H [4, 2, 0, ∞1 , 1, 5, ∞2 , 6] . (4)

Then an .H4 -decomposition of .K9 consists of the .H4 -blocks in B under the action of the map .∞i |→ ∞i and .j |→ j + 1 (mod 7). ⎞ ⎛ (4) Example 3 Let .V K10 = Z7 ∪ {∞1 , ∞2 , ∞3 } and let .B

{ = H [∞1 , 1, 2, 3, 5, 4, ∞2 , 0], H [∞3 , 1, 2, 3, 6, 4, ∞1 , 0], H [∞3 , 1, 4, 2, 5, 3, ∞2 , 0], H [∞2 , 0, 4, 1, 6, 3, ∞1 , 2], H [1, 3, 2, 0, 6, ∞3 , 4, 5], H [0, 1, 4, 2, ∞2 , ∞1 , 5, 3], H [0, 4, 3, 1, 5, ∞3 , ∞2 , 6], H [∞1 , ∞3 , 3, 1, 6, 4, ∞2 , 0], H [∞3 , 2, 4, 6, ∞1 , 3, 0, ∞2 ], } H [∞3 , 0, 4, 1, 6, 2, ∞1 , 3] . (4)

Then an .H4 -decomposition of .K10 consists of the .H4 -blocks in B under the action of the map .∞i |→ ∞i , for .i ∈ {1, 2, 3}, and .j |→ j + 1 (mod 7). ⎞ ⎛ (4) = Z11 and let Example 4 Let .V K11 { B = H [7, 8, 9, 0, 2, 3, 1, 10], H [9, 10, 4, 0, 2, 1, 5, 3], H [7, 10, 0, 1, 3, 2, 8, 5],

.

H [9, 1, 8, 0, 3, 2, 4, 6], H [10, 3, 8, 7, 1, 2, 4, 9], H [6, 7, 2, 3, 0, 8, 1, 4], H [8, 2, 5, 4, 0, 6, 1, 10], H [5, 9, 2, 4, 0, 7, 1, 8], H [7, 3, 9, 5, 0, 8, 1, 10], } H [8, 10, 2, 6, 1, 7, 3, 5] .

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Then an .H4 -decomposition of .K11 consists of the .H4 -blocks in B under the action of the map .j |→ j + 1 (mod 11). ⎞ ⎛ (4) Example 5 Let .V K12 = Z11 ∪ {∞} and let { B = H [2, 4, 1, 0, 6, ∞, 7, 3], H [4, 8, 2, 0, 1, ∞, 3, 6], H [1, 6, 3, 0, 7, ∞, 10, 9],

.

H [5, 8, 4, 0, 2, ∞, 6, 1], H [9, 10, 5, 0, 8, ∞, 2, 4], H [1, 2, 6, 0, 3, ∞, 9, 7], H [3, 6, 7, 0, 9, ∞, 5, 10], H [5, 10, 8, 0, 4, ∞, 1, 2], H [3, 7, 9, 0, 10, ∞, 8, 5], H [7, 9, 10, 0, 5, ∞, 4, 8], H [3, 7, 6, 2, 1, 10, 0, ∞], H [9, 10, 7, 1, 4, 3, 0, ∞],

} H [0, 8, 4, 1, 7, 3, 5, ∞], H [7, 9, 3, 1, 6, 5, 0, ∞], H [0, 5, 7, 1, 3, 9, 6, ∞] . (4)

Then an .H4 -decomposition of .K12 consists of the .H4 -blocks in B under the action of the map .∞ |→ ∞ and .j |→ j + 1 (mod 11). ⎞ ⎛ (4) Example 6 Let .V K15 = Z13 ∪ {∞1 , ∞2 } and let .B=

{ H [1, 2, 5, 0, ∞2 , 9, 7, 3], H [2, 4, 10, 0, 5, ∞2 , 1, 6], H [3, 6, 2, 0, ∞2 , 1, 8, 9], H [4, 8, 7, 0, 10, ∞2 , 2, 12], H [5, 10, 12, 0, ∞2 , 6, 9, 2], H [6, 12, 4, 0, ∞2 , 2, 3, 5], H [1, 7, 9, 0, 11, ∞2 , 10, 8], H [3, 8, 1, 0, 7, ∞2 , 4, 11], H [5, 9, 6, 0, ∞2 , 3, 11, 1], H [7, 10, 11, 0, 12, ∞2 , 5, 4], H [9, 11, 3, 0, ∞2 , 8, 12, 7], H [11, 12, 8, 0, ∞2 , 4, 6, 10], H [∞2 , 1, 0, ∞1 , 5, 4, 3, 2], H [∞2 , 2, 0, ∞1 , 10, 8, 6, 4], H [∞2 , 3, 0, ∞1 , 12, 2, 9, 6], H [∞2 , 4, 0, ∞1 , 7, 3, 12, 8], H [∞2 , 5, 0, ∞1 , 12, 7, 2, 10], H [∞2 , 6, 0, ∞1 , 11, 4, 5, 12], H [4, 5, 2, ∞1 , 1, 0, 3, 6], H [8, 10, 4, ∞1 , 2, 0, 6, 12], H [2, 12, 6, ∞1 , 3, 0, 9, 5], H [3, 7, 8, ∞1 , 4, 0, 12, 11], H [7, 12, 10, ∞1 , 5, 0, 2, 4], H [4, 11, 12, ∞1 , 6, 0, 5, 10], H [7, 0, ∞1 , 2, 12, 1, 4, 5], H [6, 1, ∞2 , 8, 5, ∞1 , 4, 0], H [2, 7, ∞2 , 0, 10, 1, ∞1 , 9], H [5, 1, ∞2 , 2, 7, 3, 0, 6], H [9, 12, ∞2 , 0, 6, 2, 5, 3], H [0, 1, 2, 11, 9, 8, 12, 3], H [0, 2, 4, 9, 5, 3, 11, 6], H [0, 3, 6, 7, 11, 1, 10, 9], H [0, 4, 8, 5, 10, 6, 9, 12], } H [0, 5, 10, 3, 6, 1, 8, 2], H [0, 6, 12, 1, 9, 2, 7, 5] . (4)

Then an .H4 -decomposition of .K15 consists of the .H4 -blocks in B under the action of the map .∞i |→ ∞i , for .i ∈ {1, 2}, and .j |→ j + 1 (mod 13). ⎞ ⎛ { (4) Example 7 Let .V L9,9 = .Z18 with vertex partition . {0, 2, 4, 6, 8, 10, 12, 14, 16}, } {1, 3, 5, 7, 9, 11, 13, 15, 17} and let

Spectrum of 4-Uniform 4-Colorable 3-Cycles .B

235

{ = H [0, 1, 13, 4, 3, 11, 9, 5], H [0, 5, 11, 2, 15, 1, 9, 7], H [0, 7, 1, 10, 3, 5, 9, 17], H [0, 11, 17, 8, 15, 13, 9, 1], H [0, 13, 7, 16, 3, 17, 9, 11], H [0, 17, 5, 14, 15, 7, 9, 13], H [0, 1, 10, 6, 11, 5, 9, 15], H [0, 5, 14, 12, 1, 7, 9, 3], H [0, 7, 16, 6, 5, 17, 9, 15], H [0, 11, 2, 12, 13, 1, 9, 3], H [0, 13, 4, 6, 17, 11, 9, 15], H [0, 17, 8, 12, 7, 13, 9, 3], H [0, 3, 1, 5, 6, 13, 4, 15], H [0, 15, 5, 7, 12, 11, 2, 3], H [0, 3, 7, 17, 6, 1, 10, 15], H [0, 15, 11, 1, 12, 17, 8, 3], H [0, 3, 13, 11, 6, 7, 16, 15], H [0, 15, 17, 13, 12, 5, 14, 3], H [0, 1, 15, 3, 6, 14, 4, 2], H [0, 5, 3, 15, 12, 16, 2, 10], H [0, 7, 15, 3, 6, 8, 10, 14], H [0, 11, 3, 15, 12, 10, 8, 4], H [0, 13, 15, 3, 6, 2, 16, 8], H [0, 17, 3, 15, 12, 4, 14, 16], H [1, 2, 8, 4, 3, 6, 12, 15], H [5, 10, 4, 2, 15, 12, 6, 3], H [7, 14, 2, 10, 3, 6, 12, 15], H [11, 4, 16, 8, 15, 12, 6, 3], H [13, 8, 14, 16, 3, 6, 12, 15], H [17, 16, 10, 14, 15, 12, 6, 3], H [3, 4, 11, 15, 2, 17, 10, 5], H [15, 2, 1, 3, 10, 13, 14, 7], H [3, 10, 5, 15, 14, 11, 16, 17], H [15, 8, 13, 3, 4, 7, 2, 1], H [3, 16, 17, 15, 8, 5, 4, 11], H [15, 14, 7, 3, 16, 1, 8, 13], H [0, 1, 8, 2, 5, 14, 3, 10], H [0, 5, 4, 10, 7, 16, 15, 14], H [0, 7, 2, 14, 17, 8, 3, 16], H [0, 11, 16, 4, 1, 10, 15, 2], H [0, 13, 14, 8, 11, 2, 3, 4], H [0, 17, 10, 16, 13, 4, 15, 8], H [0, 1, 7, 12, 2, 3, 13, 6], H [0, 1, 6, 7, 9, 16, 14, 5], H [0, 1, 8, 7, 14, 17, 6, 5], H [0, 2, 13, 7, 10, 12, 5, 15], H [0, 3, 8, 11, 14, 15, 10, 7], H [0, 1, 15, 4, 6, 13, 2, 3], } H [0, 1, 12, 6, 17, 10, 3, 4], H [0, 3, 16, 1, 2, 14, 8, 13] ,

{ B ' = H [0, 9, 6, 3, 7, 16, 12, 15], H [1, 10, 7, 4, 8, 17, 13, 16], H [2, 11, 8, 5, 9, 0, 14, 17], H [3, 12, 9, 6, 10, 1, 15, 0], H [4, 13, 10, 7, 11, 2, 16, 1], H [5, 14, 11, 8, 12, 3, 17, 2], H [6, 15, 12, 9, 13, 4, 0, 3], H [7, 16, 13, 10, 14, 5, 1, 4], H [8, 17, 14, 11, 15, 6, 2, 5], H [9, 0, 15, 3, 4, 13, 12, 6], H [10, 1, 16, 4, 5, 14, 13, 7], H [11, 2, 17, 5, 6, 15, 14, 8], H [12, 3, 0, 6, 7, 16, 15, 9], H [13, 4, 1, 7, 8, 17, 16, 10], H [14, 5, 2, 8, 9, 0, 17, 11], H [15, 6, 3, 9, 10, 1, 0, 12], H [16, 7, 4, 10, 11, 2, 1, 13], H [17, 8, 5, 11, 12, 3, 2, 14], H [9, 0, 17, 1, 4, 13, 10, 8], H [10, 1, 0, 2, 5, 14, 11, 9], H [11, 2, 1, 3, 6, 15, 12, 10], H [12, 3, 2, 4, 7, 16, 13, 11], H [13, 4, 3, 5, 8, 17, 14, 12], H [14, 5, 4, 6, 9, 0, 15, 13], H [15, 6, 5, 7, 10, 1, 16, 14], H [16, 7, 6, 8, 11, 2, 17, 15], H [17, 8, 7, 9, 12, 3, 0, 16], H [9, 0, 14, 4, 6, 15, 13, 5], H [10, 1, 15, 5, 7, 16, 14, 6], H [11, 2, 16, 6, 8, 17, 15, 7], H [12, 3, 17, 7, 9, 0, 16, 8], H [13, 4, 0, 8, 10, 1, 17, 9], H [14, 5, 1, 9, 11, 2, 0, 10],

} H [15, 6, 2, 10, 12, 3, 1, 11], H [16, 7, 3, 11, 13, 4, 2, 12], H [17, 8, 4, 12, 14, 5, 3, 13] .

Then an .H4 -decomposition of .L(4) 9,9 consists of the .H4 -blocks in B under the action of the map .j |→ j + 1 (mod 18) along with the .H4 -blocks in .B ' .

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⎛ (4) Example 8 Let .V L

⎞

{ = Z18 ∪ {∞} with vertex partition . {∞}, }} {0, 2, 4, 6, 8, 10, 12, 14, 16}, {1, 3, 5, 7, 9, 11, 13, 15, 17} and let 1,9,9

.B

∪L

(4) 9,9

{ = H [0, 11, 5, ∞, 1, 10, 2, 6], H [0, 1, 7, ∞, 5, 14, 10, 12], H [0, 5, 17, ∞, 7, 16, 14, 6], H [0, 13, 1, ∞, 11, 2, 4, 12], H [0, 17, 11, ∞, 13, 4, 8, 6], H [0, 7, 13, ∞, 17, 8, 16, 12], H [1, 2, 0, 4, 10, 5, 6, 3], H [5, 10, 0, 2, 14, 7, 12, 15], H [7, 14, 0, 10, 16, 17, 6, 3], H [11, 4, 0, 8, 2, 1, 12, 15], H [13, 8, 0, 16, 4, 11, 6, 3], H [17, 16, 0, 14, 8, 13, 12, 15], H [0, 11, 3, ∞, 2, 6, 1, 9], H [0, 1, 15, ∞, 10, 12, 5, 9], H [0, 5, 3, ∞, 14, 6, 7, 9], H [0, 13, 15, ∞, 4, 12, 11, 9], H [0, 17, 3, ∞, 8, 6, 13, 9], H [0, 7, 15, ∞, 16, 12, 17, 9], H [1, 2, 7, 0, 11, 10, 9, 3], H [5, 10, 17, 0, 1, 14, 9, 15], H [7, 14, 13, 0, 5, 16, 9, 3], H [11, 4, 5, 0, 13, 2, 9, 15], H [13, 8, 1, 0, 17, 4, 9, 3], H [17, 16, 11, 0, 7, 8, 9, 15], H [3, 5, 0, 1, 4, 7, 2, 9], H [15, 7, 0, 5, 2, 17, 10, 9], H [3, 17, 0, 7, 10, 13, 14, 9], H [15, 1, 0, 11, 8, 5, 4, 9], H [3, 11, 0, 13, 16, 1, 8, 9], H [15, 13, 0, 17, 14, 11, 16, 9], H [2, 11, 5, 3, 0, 8, 1, 4], H [10, 1, 7, 15, 0, 4, 5, 2], H [14, 5, 17, 3, 0, 2, 7, 10], H [4, 13, 1, 15, 0, 16, 11, 8], H [8, 17, 11, 3, 0, 14, 13, 16], H [16, 7, 13, 15, 0, 10, 17, 14], H [0, 11, 2, 5, 8, ∞, 13, 1], H [0, 1, 10, 7, 4, ∞, 11, 5], H [0, 5, 14, 17, 2, ∞, 1, 7], H [0, 13, 4, 1, 16, ∞, 17, 11], H [0, 17, 8, 11, 14, ∞, 7, 13], H [0, 7, 16, 13, 10, ∞, 5, 17], H [0, 3, 11, 1, 2, 13, 4, 6], H [0, 15, 1, 5, 10, 11, 2, 12], H [0, 3, 5, 7, 14, 1, 10, 6], H [0, 15, 13, 11, 4, 17, 8, 12], H [0, 3, 17, 13, 8, 7, 16, 6], H [0, 15, 7, 17, 16, 5, 14, 12], H [0, 1, 4, 6, 7, 12, 13, 15], H [0, 5, 2, 12, 17, 6, 11, 3], H [0, 7, 10, 6, 13, 12, 1, 15], H [0, 5, 2, 14, 12, 17, 9, 3], H [0, 1, 4, 10, 6, 7, 9, 15], H [0, 7, 10, 16, 6, 13, 9, 15], H [0, 1, 16, 3, 8, 9, 4, 5], H [0, 5, 8, 15, 4, 9, 2, 7], H [0, 7, 4, 3, 2, 9, 10, 17], H [2, 9, 7, 0, 5, 14, 1, 10], H [3, 9, 6, 0, 5, 10, 15, 12], H [1, 5, 9, 14, 0, 7, 3, 12],

} H [0, ∞, 6, 3, 9, 15, 2, 1], H [0, 3, 2, 1, 4, 13, ∞, 9], H [0, ∞, 10, 5, 3, 13, 4, 11] ,

{ B ' = H [0, 9, 12, 3, 8, 14, 2, 11], H [1, 10, 13, 4, 9, 15, 3, 12], H [2, 11, 14, 5, 10, 16, 4, 13], H [3, 12, 15, 6, 11, 17, 5, 14], H [4, 13, 16, 7, 12, 0, 6, 15], H [5, 14, 17, 8, 13, 1, 7, 16], H [6, 15, 0, 9, 14, 2, 8, 17], H [7, 16, 1, 10, 15, 3, 9, 0], H [8, 17, 2, 11, 16, 4, 10, 1], H [1, 10, 2, 11, 12, 17, 5, 14], H [2, 11, 3, 12, 13, 0, 6, 15], H [3, 12, 4, 13, 14, 1, 7, 16], H [4, 13, 5, 14, 15, 2, 8, 17], H [5, 14, 6, 15, 16, 3, 9, 0], H [6, 15, 7, 16, 17, 4, 10, 1], } H [7, 16, 8, 17, 0, 5, 11, 2], H [8, 17, 9, 0, 1, 6, 12, 3], H [9, 0, 10, 1, 2, 7, 13, 4] .

Then an .H4 -decomposition of .L(4) ∪ L(4) consists of the .H4 -blocks in B under 9,9 1,9,9 the action of the map .∞ |→ ∞ and .j |→ j + 1 (mod 18) along with the .H4 -blocks in .B ' under the action of the map .j |→ j + k (mod 18) for .k ∈ [0, 8].

Spectrum of 4-Uniform 4-Colorable 3-Cycles

237

⎞ ⎛ { (4) Example 9 Let .V L2,3,3 = Z6 ∪ {∞1 , ∞2 } with vertex partition . {∞1 , ∞2 }, } {0, 2, 4}, {1, 3, 5} and let .B

{ = H [1, 4, 5, ∞2 , 0, 3, ∞1 , 2], H [1, 4, ∞1 , 5, 0, 3, ∞2 , 2], H [0, 2, 3, ∞2 , 4, 5, ∞1 , 1], } H [∞1 , 3, 4, 5, ∞2 , 1, 0, 2], H [∞2 , 2, 1, ∞1 , 5, 0, 3, 4] . (4)

Then an .H4 -decomposition of .L2,3,3 consists of the .H4 -blocks in B under the action of the map .∞i |→ ∞i , for .i ∈ {1, 2}, and .j |→ j + k (mod 6) for .k ∈ [0, 2]. ⎞ ⎛ { (4) Example 10 Let .V L3,3,3 = Z9 with the vertex partition . {0, 3, 6}, {1, 4, 7}, } {2, 5, 8} and let { } B = H [0, 1, 2, 4, 5, 3, 6, 8], H [0, 2, 4, 8, 1, 6, 3, 7], H [0, 4, 8, 7, 2, 3, 6, 5] .

.

(4)

Then an .H4 -decomposition of .L3,3,3 consists of the .H4 -blocks in B under the action of the map .j |→ j + 1 (mod 9). ⎞ ⎛ = Z9 ∪ {∞} with vertex partition Example 11 Let .V K (4) ∪ L(4) 1,3,3,3 { } 3,3,3 . {∞}, {0, 3, 6}, {1, 4, 7}, {2, 5, 8} and let { B = H [1, 2, 0, ∞, 7, 8, 3, 5], H [0, 1, 2, 6, 4, 8, 7, 5], } H [0, 2, ∞, 4, 5, 8, 3, 1], H [0, 4, 6, 2, 7, 3, 5, 1] .

.

(4)

(4)

∪L consists of the .H4 -blocks in B under Then an .H4 -decomposition of .K 1,3,3,3 3,3,3 the action of the map .∞ |→ ∞ and .j |→ j + 1 (mod 9). ⎞ ⎛ { (4) Example 12 Let .V K2,3,3,3 = Z9 ∪ {∞1 , ∞2 } with the vertex partition . {∞1 , } ∞2 }, {0, 3, 6}, {1, 4, 7}, {2, 5, 8} and let { } B = H [0, 1, 2, ∞1 , 7, 3, 5, ∞2 ], H [0, 1, 2, ∞2 , 7, 3, 5, ∞1 ] .

.

(4)

Then an .H4 -decomposition of .K2,3,3,3 consists of the .H4 -blocks in B under the action of the map .∞i |→ ∞i , for .i ∈ {1, 2}, and .j |→ j + 1 (mod 9). ⎞ ⎛ (4) Example 13 Let .V K3,3,3,3 = Z9 ∪ {∞1 , ∞2 , ∞3 } with the vertex partition { } . {∞1 , ∞2 , ∞3 }, {0, 3, 6}, {1, 4, 7}, {2, 5, 8} and let .B

{ } = H [0, 1, 2, ∞1 , 7, 3, 5, ∞2 ], H [0, 1, 2, ∞2 , 7, 3, 5, ∞3 ], H [0, 1, 2, ∞3 , 7, 3, 5, ∞1 ] . (4)

Then an .H4 -decomposition of .K3,3,3,3 consists of the .H4 -blocks in B under the action of the map .∞i |→ ∞i , for .i ∈ {1, 2, 3}, and .j |→ j + 1 (mod 9).

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R. C. Bunge et al.

⎞ ⎛ (4) (4) = Z9 ∪ {∞1 , ∞2 , ∞3 , ∞4 , ∞5 , ∞6 } with the Example 14 Let .V K15 \ K6 .∞1 , . . . , ∞6 being the vertices in the hole and let .B

{ = H [0, 1, ∞3 , 6, 7, ∞1 , 3, ∞2 ], H [0, 2, ∞3 , 3, 5, ∞1 , 6, ∞2 ], H [0, 4, ∞3 , 6, 1, ∞1 , 3, ∞2 ], H [0, 5, ∞3 , 3, 8, ∞1 , 6, ∞2 ], H [0, 7, ∞3 , 6, 4, ∞1 , 3, ∞2 ], H [0, 8, ∞3 , 3, 2, ∞1 , 6, ∞2 ], H [0, 1, ∞4 , 6, 7, ∞6 , 3, ∞5 ], H [0, 2, ∞4 , 3, 5, ∞6 , 6, ∞5 ], H [0, 4, ∞4 , 6, 1, ∞6 , 3, ∞5 ], H [0, 5, ∞4 , 3, 8, ∞6 , 6, ∞5 ], H [0, 7, ∞4 , 6, 4, ∞6 , 3, ∞5 ], H [0, 8, ∞4 , 3, 2, ∞6 , 6, ∞5 ], H [1, 2, 0, 4, 8, 6, 3, 7], H [0, 2, 1, 5, 8, 7, 3, 4], H [0, 1, 6, 4, 7, 5, 2, 3], H [∞1 , ∞3 , ∞5 , 0, 3, 1, 4, ∞6 ], H [∞1 , 0, ∞3 , 8, ∞4 , 7, ∞2 , 1], H [∞1 , 0, ∞3 , 7, ∞4 , 5, ∞2 , 2], H [∞1 , 0, ∞3 , 5, ∞4 , 1, ∞2 , 4], H [∞4 , 0, ∞1 , 8, ∞6 , 7, ∞5 , 1], H [∞4 , 0, ∞1 , 7, ∞6 , 5, ∞5 , 2], H [∞4 , 0, ∞1 , 5, ∞6 , 1, ∞5 , 4], H [∞1 , 0, ∞5 , 8, ∞4 , 7, ∞6 , 1], H [∞1 , 0, ∞5 , 7, ∞4 , 5, ∞6 , 2], H [∞1 , 0, ∞5 , 5, ∞4 , 1, ∞6 , 4], H [∞3 , 0, ∞5 , 8, ∞2 , 7, ∞6 , 1], H [∞3 , 0, ∞5 , 7, ∞2 , 5, ∞6 , 2], H [∞3 , 0, ∞5 , 5, ∞2 , 1, ∞6 , 4], H [∞3 , 0, ∞4 , 8, ∞5 , 7, ∞2 , 1], H [∞3 , 0, ∞4 , 7, ∞5 , 5, ∞2 , 2], H [∞3 , 0, ∞4 , 5, ∞5 , 1, ∞2 , 4], H [∞1 , 0, 6, ∞3 , 1, 4, ∞2 , 3], H [∞1 , 0, 6, ∞4 , 1, 4, ∞5 , 3], H [∞3 , 0, 6, ∞4 , 1, 4, ∞6 , 3], H [∞5 , 0, 6, ∞2 , 1, 4, ∞6 , 3], H [∞4 , ∞6 , 0, ∞1 , 2, ∞3 , ∞2 , 1], H [∞4 , ∞5 , 0, ∞2 , 2, ∞6 , ∞3 , 1], H [∞5 , ∞6 , 0, ∞2 , 2, ∞4 , ∞3 , 1], H [∞1 , ∞2 , 0, ∞4 , 2, ∞3 , ∞6 , 1], H [2, 4, 6, ∞1 , 8, 5, ∞6 , 0], H [2, 4, 6, ∞3 , 8, 5, ∞5 , 0], H [2, 4, 6, ∞2 , 8, 5, ∞4 , 0], H [0, 1, 2, ∞1 , 3, ∞5 , ∞2 , 5], H [0, 1, 2, ∞2 , 3, ∞5 , ∞3 , 5], H [0, 1, 2, ∞3 , 3, ∞1 , ∞4 , 5], H [0, 1, 2, ∞4 , 3, ∞1 , ∞5 , 5], H [0, 1, 2, ∞5 , 3, ∞4 , ∞6 , 5], } H [0, 1, 2, ∞6 , 3, ∞5 , ∞1 , 5] ,

{ B ' = H [0, 3, ∞1 , 6, 1, 7, 4, 8], H [1, 4, ∞1 , 7, 2, 8, 5, 0], H [2, 5, ∞1 , 8, 3, 0, 6, 1], H [3, 6, ∞2 , 0, 1, 4, 7, 2], H [4, 7, ∞2 , 1, 2, 5, 8, 3], H [5, 8, ∞2 , 2, 3, 6, 0, 4], H [0, 6, ∞3 , 3, 4, 7, 1, 5], H [1, 7, ∞3 , 4, 5, 8, 2, 6], H [2, 8, ∞3 , 5, 6, 0, 3, 7], H [0, 3, ∞4 , 6, 2, 8, 5, 7], H [1, 4, ∞4 , 7, 3, 0, 6, 8], H [2, 5, ∞4 , 8, 4, 1, 7, 0], H [3, 6, ∞5 , 0, 2, 5, 8, 1], H [4, 7, ∞5 , 1, 3, 6, 0, 2], H [5, 8, ∞5 , 2, 4, 7, 1, 3], } H [0, 6, ∞6 , 3, 5, 8, 2, 4], H [1, 7, ∞6 , 4, 6, 0, 3, 5], H [2, 8, ∞6 , 5, 7, 1, 4, 6] .

Spectrum of 4-Uniform 4-Colorable 3-Cycles (4)

239 (4)

Then an .H4 -decomposition of .K15 \ K6 consists of the .H4 -blocks in B under the action of the map .∞i |→ ∞i , for .i ∈ {1, 2, 3, 4, 5, 6}, and .j |→ j + 1 (mod 9) along with the .H4 -blocks in .B ' .

3 Main Constructions The constructions in this section are dependent on the many small examples given in Sect. 2. We proceed by proving a lemma that is an extension of an approach in [6] and is fundamental to our constructions. Lemma 3 Let .n ≥ 1, .x ≥ 0, and .r ≥ 0 be integers and let .v = nx + r. There exists (4) a decomposition of .Kv into the following: (4)

• 1 copy of .Kn+r ,

.

(4) (4) • .x − 1 copies of .Kn+r \ Kr(4) (these are isomorphic to .Kn+r if .r ∈ [0, 3]), (x ) (4) (4) (4) .• . 2 copies of .Lr,n,n ∪ Ln,n (here .Lr,n,n is empty if .r = 0), (x ) (4) (4) (4) .• . 3 copies of .Kr,n,n,n ∪ Ln,n,n (here .Kr,n,n,n is empty if .r = 0), and (x ) (4) .• . 4 copies of .Kn,n,n,n . .

Proof If .x ∈ {0, 1}, the decomposition is trivial. Thus we may assume that .x ≥ 2. Let .V0 , V1 , . . . , Vx be pairwise disjoint sets of vertices with .|V0 | = r and .|V1 | = (4) |V2 | = · · · = |Vx | = n and let .V = V0 ∪ V1 ∪ · · · ∪ Vx . Then, .KV can be viewed as the (edge-disjoint) union (4)

KV1 ∪V0 ∪

.

| | ⎛ 2≤i≤x

∪

(4)

(4)

KVi ∪V0 \ KV0

| |

⎛

(4)

⎞

| |

∪

1≤i 3, the analysis in [4] should transfer to this case and show not only that there is an upper bound (depending on .⎿z⏌) for the number of intervals in a state, but what the states are. 2. The patterns which hold for .z > 3 continued backwards to the case .1 < z < 3 should give legitimate tilings (though not the greedy ones). 3. It would be rather interesting if this did ever happen. I would look for this phenomenon in some situations when the continued fraction of .z < 3 has denominators bounded by .2. 4. Even in the event this did happen, the alternate point of view given before the proof of Theorem 6 might get past the objection. Instead of imagining a sequence of blocks .C ⊕ [u, v) taken one at a time there might be continuously indexed tiles .Tt and situations .Vt with first empty space .αt where .t ≥ 0 and .αt is piece-wise linear of slope 1 with jump discontinuities. Since the measure of the non-negative portion of .Vt (the union of the tiles to that point) is 3t one should be able to get .3t − 1 − z ≤ αt ≤ 3t. These concerns could perhaps be dealt with more effectively by using a modification of the greedy rule as mentioned at the end of Sect. 5.2. Let .y = min(1, z−1). Then, in a situation with first empty space .α, any currently empty space in .[α, α + 1) can now only be filled by the initial point of a tile. The greedy rule could then specify “fill all the empty spaces in the interval .[α, α + y) with the start of a tile using the tile of type B only for those spaces where the tile type A cannot.” As noted in Claim 2, for any empty space .x ' ∈ [α, α + y), the possible choices for the tile starting at .x ' are unaffected by the choices made for empty spaces .x ∈ [α, x ' ). The proof of Theorem 6 would need to be adjusted to this modified version of the greedy rule. This can be done and has the advantage that it establishes: Claim 4 If .S0 is any brief state then the modified greedy rule starting at .S0 tiles the rest of .R. This allows .S0 to be any subset of .(0, z) provided that the definition of a situation is appropriately broadened.

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5.4 Interval Tilings Theorem 6 leaves open the question: Can a bounded interval be tiled by .{0, 1, 1 + z} and .{0, z, 1+z} for z irrational? It is not possible to do an exhaustive search as in the integer case. There are (uncountably) many choices for .z, so we need to somehow consider many cases at the same time. This restriction leads to Theorem 8 which provides an affirmative answer and, in the integer case, improves the upper bound for .sab . Consider all the z with integer part .⎿z⏌ = p so .z = p + λ with .0 ≤ λ < 1. One might look for tilings using only translates of blocks of the 6 forms .C ⊕ [0, 1), C ⊕ [0, λ) and .C ⊕ [0, 1 − λ) such that the tilings are simultaneously valid for all .0 ≤ λ < 1. This proves to be very fruitful with patterns which depend on the congruence class of .p mod 3. It turns out a bit nicer to allow .0 ≤ λ ≤ 1. The options .λ = 0 and .λ = 1 are only available when z is an integer. As discussed following Theorem 6, this provides two different tilings. At these transition points where one tiling pattern ends and another starts, both are valid. The situation is particularly simple when .⎿x⏌ mod 3 = −1. Theorem 7 Let j be a positive integer and .z = (3j − 1) + λ with .0 ≤ λ ≤ 1. Consider .A = {0, 1, 1 + z} and .B = {0, z, 1 + z}. Then translates of these two prototiles tile the interval .[0, 6j ). Proof We have A = {0, 1, 3j + λ} and B = {0, 3j − 1 + λ, 3j + λ}

.

Then A ⊕ [0, 1 − λ) = [0, 1 − λ) ∪ [1, 2 − λ) ∪ [3j + λ, 3j + 1)

.

and B ⊕ [1 − λ, 1) = [1 − λ, 1) ∪ [3j, 3j + λ) ∪ [3j + 1, 3j + 1 + λ)

.

are disjoint sets whose union is [0, 2 − λ) ∪ [3j, 3j + λ + 1).

.

Similarly, (A ⊕ [2 − λ, 2)) ∪ (B ⊕ [2, 3 − λ)) = [2 − λ, 3) ∪ [3j + λ + 1, 3j + 3).

.

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347

Together these give a tiling of .[0, 3) ∪ [3j, 3j + 3). String j of these together to get a tiling of .[0, 6j ) .█ We might describe this tiling by the word .

⎛ ⎞j A1−λ B λ Aλ B 1−λ .

It was then a matter of carefully verifying that this a legitimate tiling. The next theorem covers all real .z > 1. The proof consists of explicitly described tilings. These depend on the congruence class . mod 3 of .⎿z⏌. In Theorem 7 we gave the easiest case, .⎿z⏌ ≡ −1 mod 3. The other two cases are similar, but the tilings, not given here, have longer descriptions. Theorem 8 Let .A = {0, 1, 1 + z} and .B = {0, z, 1 + z} with .z = p + λ , .p ≥ 1 an integer and .0 ≤ λ ≤ 1. Then there is a tiling by translates of these two sets of the bounded interval .[0, 3m) where ⎧ ⎪ ⎪ ⎨4j + 4 + 2λ z = 3j − 2 + λ .m = 2j z = 3j − 1 + λ . ⎪ ⎪ ⎩4j + 2 + 2λ z = 3j + λ Furthermore, this tiling uses only translates of .C ⊕ [0, 1), C ⊕ [0, λ) and .C ⊕ [0, 1 − λ) for .C = A and B. For .z = ab rational, these scale up so .z = p + λ becomes the usual division with remainder .b = pa + r with .0 ≤ r < a. This gives Theorem 6. In [3] there are results of this nature.

6 Further Results and Questions Regarding Dab In this optional section we mention results and open problems concerning these tilings. For proofs, further results, details, and other open problems, see [5]. For the reasons mentioned above, we assume that .0 < a < b and .gcd(a, b) = 1. Hence, if .a ≡ b mod 3 then .{0, a, n} is a complete set of residues .mod 3 as is .{0, b, n}. When this is not the case, as in .D23 , it appears that .Dab has a single connected component containing all the cycles. When it is the case, as in .D14 , the situation is quite different. Here is .D14 represented using the short form. Each connected component is labeled with a weight function defined in Theorem 9.

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One can see from .D23 that – Every tiling of an interval, and every cycle in the graph, uses an even number of tiles, half of each type. – The intervals which can be tiled are all those of length 6t for .t > 2. – All cycles are in the same strongly connected component. – A tiling of .N or .Z uses both kinds of tiles. On the other hand, .D14 has six connected components, two of size 1 and the other four containing cycles. ABA and BAB describe tilings of .[0, 8), Setting .Tj = {3j, 3j + 1, 3j + 5} for .j ∈ Z is a tiling of the integers using only type A. In .D14 there is a loop (labelled A) from the state .{0, 1, 2, 5} to itself and the walk corresponding to this tiling repeats this loop endlessly. It turns out that the intervals which can be tiled are those of length 3t for .t > 2. Theorem 9 If .a ≡ b mod 3, then there are tilings of .Z using only one type of tile. Also, .Dab typically has many connected components containing cycles. Proof Let .C = A or .B and set .Tj = C +3j for .j ∈ Z. This is a tiling of the integers. Let .ω = e

2π i 3

so .1 + ω + ω2 = 0. Consider this weight function on finite sets .S ⊂ N:

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.

wt(S) =

⎲

ωi

i∈S '

where .S ' = S, {0} ∪ (S + 1) or .{0, 1} ∪ (S + 2) according as .|S| mod 3 = 0, 2 or .1, so .|S ' | mod 3 = 0. Note that .wt(T ) = 0 for a tile. When we adjust .Si' ∪ Ti+1 to get ' .S i+1 , we drop 3s consecutive full spaces for some .s ≥ 0, so the weight is preserved. Accordingly, .wt(S) is invariant for the states in each connected component of .Dab . Recall that a state is brief if the distance from the first empty space to the last full space is less than .b. Then the greedy rule starting at .S0 = S creates a cycle. The various weights of the brief states include all .p0 + p1 ω + p2 ω2 with .0 ≤ pi < b3 and .p0 + p1 + p2 ≡ 0 mod 3. Taking into account that .1 + ω + ω2 = 0, we can also require that at least one of .p0 , p1 , p2 is .0. Together this means that there are roughly b 2 .( ) different weights for connected components with cycles. This is a rather weak 3 bound, since there appear to be distinct connected components with the same weight for .a + b > 8. .█ In this case that .{0, a, a + b} is a complete set of residues . mod 3 we have Problem 3 Is there a constant .k = kab so that every interval of length 3t for .t > k has a tiling? In the other case we have: Problem 4 Suppose .a /≡ b mod 3, prove or disprove: 1. All cycles have even length and involve an equal number of tiles of each type. 2. All the cycles belong to the same connected component of .Dab . 3. There is a constant .k = kab so that every interval of length 6t for .t > k has a tiling. We provide the following remarks for each statement in Problem 4. 1. This is equivalent to saying that there is a height function .ht on the states so that for an edge from .S1 to .S2 , .

ht(S2 ) = ht(S1 ) ± 1

according as the edge is type A or B. So it would suffice to show how to compute ht(S) from S alone. 2. A quarter of the states are dead ends and can not belong to a cycle. Of the rest, one third have two edges leaving them and the rest have one. Similar remarks hold for incoming edges. This creates many possibilities for connections between states in cycles. 3. Every brief state belongs to a greedy cycle which is relatively short. If the component of .[0, n + 1) contains two cycles of lengths u and v with .gcd(u, v) = 2, then the result follows. In practice there seem to be cycles of many lengths. .

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References ´ 1. Sands, A. D. and Swierczkowski, S. Decomposition of the line in isometric three-point sets. Fund. Math. 48 (1959/60) 351–362. 2. Gordon, B. Tilings of lattice points in Euclidean n-space. Discrete Math 29 (1980) 169–174. 3. Adler, A. and Holroyd, F.C. Some results on one-dimensional tilings. Geom. Dedicata 10 (1981) 49–58. 4. Meyerowitz, A. Tilings in Z with triples. J. Comb. Theory Ser. A 48 (1988) 229–235. 5. Meyerowitz, A. Tiling the line with triples. Discrete models: combinatorics, computation, and geometry (Paris, 2001), Discrete Math. Theor. Comput. Sci. Proc., AA (2001) 257–274.

Deques on a Torus Thomas McKenzie and Shannon Overbay

Abstract We provide an overview of graphical representations of stacks, queues, and deques. In the case of a cylinder book, corresponding to a deque, we give edge bounds and determine the deque number of the complete graph. We extend the deque in a natural way, forming a toroidal deque. Unlike the other three structures, the toroidal deque can process certain non-planar graphs, including .K7 and the Cartesian product of two cycles. Keywords Book thickness · Queue · Stack · Deque

1 Introduction Stacks, queues, and deques are commonly used data structures. Such structures can be visualized with graphs that can be drawn in the plane in various ways without edge crossings. In the stack case, this can be achieved with a circular layout of the vertices that corresponds to a book drawing (see [3, 5], and [8]). A path layout of the vertices with edges wrapping around the ends can be used to represent a queue (see [6–8], and [10]). A deque can be represented by a cylindrical book page with the vertices in a linear arrangement and non-crossing edges on the side of the cylinder (see [1, 2, 13], and [15]). Section 2 provides several known results for these structures. In Sect. 3, we extend cylinder embeddings to the torus. The deque contains the queue and the stack, but is still a planar structure. By joining the ends of the cylinder we form a torus book page containing a circular layout of the vertices. The corresponding toroidal deque allows us to process non-planar graphs, such as a .K7 . Section 4 includes edge bounds and optimal embeddings of complete graphs. The results for cylinder books are new. In Sect. 5 we provide an overview of embeddings of Cartesian products of paths and cycles on these structures. We

T. McKenzie (✉) · S. Overbay Gonzaga University, Spokane, WA, USA e-mail: [email protected]; [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 S. Heuss et al. (eds.), Combinatorics, Graph Theory and Computing, Springer Proceedings in Mathematics & Statistics 462, https://doi.org/10.1007/978-3-031-62166-6_25

351

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extend the results to toroidal deques in the case of the product of two cycles in a manner that preserves the relative vertex ordering of the original cycles.

2 Background We first consider a structure called a book (see [3] and [11]). An n-book consists of n half planes joined together at a common line, called the spine. In a book embedding the vertices of the graph are placed in a linear order on the spine and each edge of the graph is assigned to a single page so that no two edges cross each other or the spine. The book thickness of a graph G is the smallest n for which G has an n-book embedding. Figure 1 gives a one page embedding of a graph. If the linear order of the vertices along the spine is .v1 , v2 , . . . , vm , we may add any missing edges of the form .{vk , vk+1 }, k = 1, 2, . . . , m − 1, or the edge .{v1 , vm } onto any page of the book without creating edge crossings. Bernhart and Kainen (Theorem 2.5, p. 322 [3]) gave the following useful characterizations of one and two page book embeddings. Theorem 1 Let G be a graph. 1. The book thickness of G is less than or equal to one if and only if G is outerplanar. 2. The book thickness of G is less than or equal to two if and only if G is a subgraph of a planar graph with a Hamiltonian cycle. Given a one page book embedding, we can arrange the vertices in a circle. The edges can then be realized as non crossing chords of this circle. Figure 1 (right) demonstrates how to do this. Determining book thickness with respect to the circular ordering is equivalent to finding the minimum number of colors needed to color the chords of the circle so that no two chords of the same color cross. Each page of a book corresponds to a one-sided stack. A one-sided stack is a linear data structure in which items are added and removed from one side following a last-in, first-out convention. If .i < j and a book embedding has an edge connecting .vi to .vj , we view this as adding the edge connecting .vi to .vj to the stack at step i and removing it at step j . If more than one edge is added at step i, the edges are

Fig. 1 Two embeddings of the same graph

Deques on a Torus

353

Table 1 Stack corresponding to Fig. 1 Stack

Step 1 2 3 4 5 6

Action Add .e5 to stack, then add .e1 Add .e3 to stack, then add .e2 Remove .e2 Remove .e3 , then remove .e1 . Now add .e6 followed by .e4 Remove .e4 Remove .e6 , then remove .e5

.e5 , e1 .e5 , e1 , e3 , e2 .e5 , e1 , e3 .e5 , e6 , e4 .e5 , e6 .∅

Fig. 2 One page cylinder book embedding

1

2

3

4

5

6

Fig. 3 Two representations of the same embedding

added in the reverse order of the corresponding indices. The fact that edges do not cross in a book embedding assures us that the edge from .vi to .vj will be at the top of the stack (hence not obstructed) at step j [5]. Table 1 shows how this stack works with the graph in Fig. 1. In [13] and [12], Overbay defined a cylinder book. In such a book, vertices are placed in a line parallel to the axis of the cylinder. The pages of a cylinder book are cylinders, rather than half-planes, joined together at the spine. This can be represented in the plane as a square with the spine along the top and bottom of the square. Figures 2 and 3 show one-page cylinder embeddings of graphs. In Fig. 3, on the left we display the graph on the cylinder and to the right we display it on the

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T. McKenzie and S. Overbay

Table 2 Queue corresponding to Fig. 2 Step 1 2 3 4 5 6

Queue .e1 , e2 .e1 , e2 , e3 .e2 , e3 , e4 , e5 .e5 , e6 , e7 .e7 , e8 .∅

Action Add .e1 right, then add .e2 right Add .e3 right Remove .e1 . Add .e4 right, then .e5 right Remove .e2 , .e3 , and .e4 . Add .e6 right, then .e7 right Remove .e5 and then .e6 . Add .e8 right Remove .e7 , then remove .e8

unrolled cylinder. Note that in Fig. 2, all edges wrap around the cylinder from left to right. Auer, et al., noted that this linear cylindrical embedding describes a one-sided queue [1]. A one-sided queue is a linear data structure in which items are added to one side of the queue and removed from the other side following a first-in, first-out convention. If .i < j and a linear cylindrical embedding has an edge from .vi to .vj , we view this as adding the edge connecting .vi to .vj to the right side of the queue at step i and removing it from the left at step j . If more than one edge is added at step i, the edges are added in the order of the corresponding indices. Again, the fact that edges do not cross ensures us that an edge will not be obstructed by a different edge when it is time to take it off the queue [1]. Table 2 shows how to process the queue corresponding to the graph in Fig. 2. Stacks and queues are generally not comparable in that they admit different types of edges [10]. We note that Dujmovic et al. have compared stacks and queues in [8] and have made significant recent advancements in this area (see [6] and [7]). Next we consider a combination of stacks and queues in a data structure called a deque. Elements can be placed on either side of a deque and then can be removed from either side. One page cylinder book embeddings correspond to a deque [1]. Overbay [13] and Auer [2] independently gave the following useful characterization of a one page cylinder book. Theorem 2 A graph G is embeddable in a one page cylinder book if and only if it is a subgraph of a planar graph with a Hamiltonian path. Since there exist maximal planar graphs which have Hamiltonian paths but not Hamiltonian cycles [3], we see that a two page book is properly contained in a one page cylinder book. We process a deque using the conventions of Auer, et al. Assume .i < j . If an edge connects .vi to .vj by wrapping around the cylinder from left to right, we put that edge on the right side of the deque at time i and remove it from the left at time j . Edges which connect .vi to .vj by wrapping around the cylinder from right to left are added to the left side of the deque at time i and removed from the right at time j . If an edge connects .vi to .vj below the spine, we put that edge on the right side of the deque at time i and remove it from the right at time j . Finally, edges which connect .vi to .vj above the spine, are placed on the left side of the deque at time i

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355

Table 3 Deque corresponding to Fig. 3 Step 1 2 3 4 5 6

Deque .e1 , e2 .e1 , e2 , e3 , e4 .e2 , e3 , e4 , e5 .e7 , e6 , e4 .e8 , e7 .∅

Action Add .e1 left, then add .e2 right Add .e3 right, then .e4 right Remove .e1 . Add .e5 right Remove .e2 , .e3 , and .e5 . Add .e6 left followed by .e7 left Remove .e4 and then .e6 . Add .e8 left Remove .e7 right, then remove .e8 left

and removed from the left at time j . Table 3 shows how to process the deque graph from Fig. 3. One way to view a stack (queue, deque, resp.) is as a collection of documents. The ordered vertices can be thought of as offices in a hallway and the edges as documents. If .i < j and .vi is connected to .vj by edge e, then the occupants in offices .vi and .vj need to sign document e. An assistant walks down the hall from office .v1 to .vn to collect the signatures. When the assistant arrives at office .vi , the occupant in that office signs document e, and document e is placed on the top (bottom, top or bottom, resp.) of the stack (queue, deque, resp.). When the assistant arrives at office .vj , document e is removed from the top (bottom, top or bottom, resp.) of the stack (queue, deque, resp.), the occupant in office .vj sign it, and it is then placed in a different, completed, pile. Order does not matter in this completed pile. In Fig. 1, on the left, the assistant arrives at office .v4 with the stack of papers .e5 , e1 , e3 . Documents .e3 and .e1 are removed from the top of the stack and are signed by the occupant in office .v4 . Documents .e3 and .e1 are placed in the completed pile. The occupant in office .v4 then signs documents .e4 and .e6 , which are placed on the top of the stack in the order .e6 followed by .e4 . Now the assistant moves on to office .v5 . One of the advantages of viewing the situation in this manner is that if two people need to sign one document, our intuition tells us it does not matter who signs the document first. This will be helpful in the next section.

3 Toroidal Deques One stack, two stacks, queues, and deques are all planar structures. In [13] and [12], Overbay defined a non-planar structure called a torus book. Torus books are similar to cylinder books, but vertices are placed in order on a circular spine (an equator of the torus) rather than a line, and pages are tori joined together at the spine, rather than cylinders. This can be viewed as forming a cylinder book and then gluing together the ends of each cylindrical page. So a torus page clearly contains a cylinder page. In the next section we will show this containment is proper. Again, the goal is to place all edges on the torus pages without crossing edges and without crossing the spine. The minimum number of torus pages need to achieve this is called the

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Fig. 4 One page torus book embedding

Fig. 5 A different representation of the graph in Fig. 4

Table 4 Toroidal deque corresponding to Fig. 5

Step 1 2 3 4 5 6 7 8 9

Toroidal deque .e1 , e2 .e1 , e3 .e5 , e4 , e3 .e4 .e8 .e8 , e7 , e6 .e8 , e7 .e7 .∅

Action Add .e1 right, then .e2 right Remove .e2 right, add .e3 right Remove .e1 . Add .e4 left, .e5 left Remove .e3 and .e5 Remove .e4 right, add .e8 left Add .e7 and .e6 right Remove .e6 Remove .e8 Remove .e7 left

torus book thickness of the graph. Figure 4 gives a one-page torus embedding of a graph. It is possible to interpret torus book embeddings as deques. To see this, we have redrawn the Fig. 4 torus book embedding in Fig. 5. This new picture has the advantage that it looks like our pictures of cylinder book embeddings, so the edges can be processed from left to right using the same convention. Table 4 shows how to interpret this embedding as a toroidal deque. Note that Auer developed a similar concept called a cyclic deque in [2]. It is always possible to represent a one page torus book embedding as a parallelogram as in Fig. 6. If we have no edge that wraps from top to bottom, then

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Fig. 6 Torus embedding as a parallelogram

we have a two page book. If there is an edge e from top to bottom, without loss of generality .vi to .vj where .i < j , this edge forms the left and right boundaries of the parallelogram. The top and bottom are bounded by the spine. This parallelogram contains all edges of the one page torus book embedding without crossings (see Fig. 6). In any such depiction we can always add the n edges connecting consecutive vertices on the top of the parallelogram. In this way we see such graphs are subgraphs of toroidal Hamiltonian graphs where the Hamiltonian cycle coincides with the equator of the torus. Looking at Fig. 6, we see that at step 1 we process .vi and we only work on the right side of the deque for the next .j − i − 1 steps through .vj −1 . Now we proceed from steps .j − i through n processing .vj through .vi working on both sides of the deque. Finally, on steps .n + 1 through .n + 1 + (j − i) we process .vi through .vj working only on the left side of the deque. Notice that this leaves the right side of the deque open during these final steps. If .i > j in our scenario, just reverse right and left in the description above. The deque is processed in .n + 1 + (j − i) ≤ 2n steps. A toroidal deque can be thought of as signing documents, but in this case the offices are arranged in a circular hall. In Table 4, the assistant starts at office .v1 , makes one complete loop, and ends at office .v3 . For example, the assistant leaves office .v6 with the deque of documents .e8 , e7 , e6 . At step seven, the assistant loops back to office .v1 , has the occupant sign document .e6 (which has already been signed by the occupant of office .v6 ), places it in the completed file, and moves to office .v2 .

4 Edge Bounds In this section we consider bounds for the number of edges which can be placed on k page books (stacks), queues, cylinder books (deques), and torus books (toroidal deques). Then we specifically consider how complete graphs fit on these objects. We note here that the results for cylinder books (deques) are new. The edge bound for books was given by Bernhart and Kainen in [3]. Theorem 3 Let e be the number of edges, k be the number of pages (stack number), and n be the number of vertices in a book embedding of a graph. Then

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e ≤ n + k(n − 3).

.

The queue number of a graph is analogous to its book thickness. It is the minimum number of queues necessary to process all edges of the graph with respect to a common linear ordering. A concise proof of the following bound for the queue number of a graph can be found in [8]. Theorem 4 Let e be the number of edges, k be the queue number, and n be the number of vertices in a graph. Then e ≤ 2kn − k(2k + 1).

.

(1)

Below, we prove a new result for edge bounds on a k page cylinder book (deque). Theorem 5 Let e be the number of edges, k be the number of cylinder pages (deque number), and n be the number of vertices in a cylinder book embedding of a graph. Then e ≤ (n − 1) + k(2n − 5).

.

(2)

Proof To obtain this bound, it is helpful to represent the cylinder page as a circle as in Fig. 7. Since there is no edge from the first and last vertices to themselves, we only need to place them in the circle once. Of course, each edge on the outside of the circle appears on at most one page, but we temporarily place them all on every page under consideration for ease of explanation. This circular ordering has .2n − 2 vertices. We maximize the number of edges in the interior this circular arrangement by forming triangles as in Fig. 7. This is known as a triangulation. Any triangulation will have .(2n − 2) − 3 = 2n − 5 interior edges. Counting the .n − 1 edges around the circle, we see that a k-page cylinder book has at most .(n − 1) + k(2n − 5) edges. Fig. 7 Cylinder page as a circle

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Finally, the edge bound for torus books (toroidal deques) can be found in [13] and [12]. Theorem 6 Let e be the number of edges, k be the number of torus pages (toroidal deque number), and n be the number of vertices in a torus book embedding of a graph. Then e ≤ n + 2kn.

(3)

.

Next, we show that equality can hold in Theorem 5. Theorem 7 There exists a graph which can be embedded in a two page cylinder book such that e = (n − 1) + 2(2n − 5)

(4)

.

where e is the number of edges and n is the number of vertices of the graph. Proof Figure 8 gives such a graph. It has 39 edges, 10 vertices, and is embedded on a two-page cylinder book. A cylinder book clearly contains a two-page book, and thus a two-cylinder book contains a four-page book. Theorem 3 implies that a four-page book has no more than .5n − 12 edges and Theorem 5 implies a two page cylinder book has no more than .5n − 11 edges, so it is natural to ask if there exists a graph which fits on a two-page cylinder, but not a four-page book. Corollary 1 There exists a graph which can be embedded in a two page cylinder book but cannot be embedded in a four-page book. Fig. 8 Maximal two page cylinder book embedding

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Proof The graph in Fig. 8 is embedded in a two-page cylinder book. Since this graph has 10 vertices and 39 edges, and since .39 > 10 + 4(10 − 3), Theorem 3 implies it cannot be embedded in a four-page book. Now that we have edge bounds, we consider how many pages it takes to embed a complete graph, .Kn , on books, queue layouts, cylinder books, and torus books. Below we include new results in Part 3 which show that the deque number of .Kn is at least .⎾n/4⏋. This extends known results that the deque number is at most .⎾n/4⏋. Theorem 8 Let .Kn be the complete graph on n vertices. Then 1. 2. 3. 4.

The stack number of .Kn is .⎾n/2⏋ (the ceiling of .n/2) The queue number of .Kn is .⎿n/2⏌ (the floor of .n/2) The deque number of .Kn is .⎾n/4⏋ The toroidal deque number of .Kn is .⎿n/4⏌

Proof The proof of 1 can be found in [3], the proof of 2 can be found in [10], and the proof of 4 can be found [13]. To prove 3, by the division algorithm, .n = 4t + 1, 4t + 2, 4t + 3, or .4t + 4. Since every cylinder page contains two book pages, by part 1 of this theorem, the cylinder book thickness of .Kn is less than or equal to .⎾n/4⏋ = t + 1. Now suppose by way of contradiction that the cylinder book thickness is less than or equal to t. If we prove this cannot happen in the case where .n = 4t +1, the other three cases will follow. Now, if .n = 4t +1, then by Theorem 5,

.

t≥ =

(n(n − 1)/2) − n + 1 e−n+1 = 2n − 5 2n − 5

(5)

((4t + 1)(4t)/2) − (4t + 1) + 1 − 2t = . 2(4t + 1) − 5 8t − 3 8t 2

Multiplying both sides by .8t − 3 yields .8t 2 − 2t ≤ t (8t − 3). But this implies .−2 ≤ −3 which is a contradiction. Thus, the cylinder book thickness of .Kn is .⎾n/4⏋. The previous theorem illustrates the added value of the torus book. In particular, the non-planar graph .K7 has stack number 4, queue number 3, deque number 2, and toroidal deque number 1. Figure 9 gives an embedding of .K7 in a one page torus book. We sum up the proper containments established so far below. one stack ⊂ two stacks ⊂ one deque ⊂ one toroidal deque

.

1-book ⊂ 2-book ⊂ one page cylinder book ⊂ one page torus book

.

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Fig. 9 .K7 on one page torus

5 Cartesian Products In this section we consider embeddings of Cartesian products of graphs (also see page 22 of [9]). If .G1 and .G2 are graphs, we write .G1 □G2 for the Cartesian product of .G1 and .G2 . Recall that the vertices of .G1 □G2 are the elements .(g1 , g2 ) ∈ G1 × G2 and that .(g1 , g2 ) is adjacent to .(h1 , h2 ) when 1. .g1 = h1 and .g2 is adjacent to .h2 or 2. .g2 = h2 and .g1 is adjacent to .h1 . The stack and queue numbers for Cartesian products of graphs have been studied in [3, 4, 10, 13, 14], and [16]. Write .Pn for the path with n vertices and .Cn for the cycle with n vertices. The following theorem collects some of the known results for products of paths and cycles and frames them in terms of cylinder book thickness. Theorem 9 Let .m, n ∈ N − {0}. 1. The queue number of .Pm □Pn is one. 2. If .m > 2 and .n > 2, the book thickness of .Pm □Pn is two. 3. If .m > 2, .Pm □Cn has book thickness two. Hence in all three cases, the cylinder book thickness is one. Proof The proof of part 1 follows from Theorem 4.2 on page 947 of [10]. To prove part 2, first note that .Pm □Pn contains a subgraph homeomorphic to .K2,3 , so by Theorem 11.10 on page 107 of [9], it is not outerplanar. However, it is easy to see that .Pm □Pn is subhamiltonian, and hence its book thickness is two. To prove part 3, again note that .Pm □Cn contains a subgraph homeomorphic to .K2,3 , so it is not outerplanar. But it is subhamiltonian so its book thickness is two. In parts 2 and 3 of the last result, the Cartesian products are embedded in books by tracing a Hamiltonian path and using it to induce a linear vertex ordering on the spine. The problem is that this Hamiltonian path might not preserve the order of the vertices in either of the two graphs which make up the Cartesian product. Additionally, some proofs of the above results use alternating vertex orderings for each copy of one of the two graphs (forward and reverse) along the spine (see, for example, [3]). We now give alternate one page cylinder embeddings which preserve the order and maintain the orientation of the vertices of both original graphs.

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First we consider .Cm □Cn , where .m > 2 and .n > 2. Such a graph has a subgraph homeomorphic to .K5 , so by Kuratowski’s Theorem (Theorem 11.13 page 109 of [9]), it is not planar. Thus it can not be embedded on one cylinder page. However, we can embed it on a one page torus book in a manner which preserves the ordering of the vertices of the original two cycles which make up .Cm □Cn . Theorem 10 For all .n, m ∈ N, .Cm □Cn can be embedded in one torus page in a manner which respects the order of the vertices of the original graphs. Proof Figure 10 shows how to do this. Now we can revisit embedding .Pm □Pn and .Pm □Cn in cylinder books. Theorem 11 For all .n, m ∈ N, both .Pm □Pn and .Pm □Cn can be embedded in one page cylinder books in a manner which respects the order of the vertices of the original graphs. Proof If we remove the edges that wrap horizontally around the torus, for example the edges passing through A, B, C, and D in Fig. 10, we have .Pm □Pn on a cylinder in a manner which respects the order of the vertices of the original graphs. Since we can embed .Pm □Cn in one cylinder page, we can clearly embed .Pm □Pn in one cylinder page. The reader should work though the document signing example with our embeddings of these cross products and notice how the order of the offices is preserved.

References 1. Auer C., Bachmaier, C., Brandenburg, F.J., Brunne,r W.; Gleißner, A.: Data structures and their planar layouts, Journal of Graph Algorithms and Applications, 22 (2), 207–237 (2018) 2. Auer, C.: Planar graphs and their duals on cylinder surfaces, Ph.D. Dissertation, University of Passau, (2014)

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3. Bernhart, F., Kainen, P. C.: The book thickness of a graph, Journal of Combinatorial Theory, Series B, 27 (3), 320–331 (1979) 4. Chen, Y.: Layout of planar products, J. Math. Comput. Sci., 6, 216–229 (2016) 5. Chung, F. R. K., Leighton, F. T., Rosenberg, A. L.: Embedding graphs in books: A layout problem with applications to VLSI design, SIAM Journal on Algebraic and Discrete Methods, 8 (1), 33–58 (1987) 6. Dujmovic, V., Eppstein, D., Hickingbotham, R., Morin, R., Wood, D. R.: Stack-Number is Not Bounded by Queue-Number. Combinatorica, 42, 151–164 (2022) 7. Dujmovic, V.; Joret, G.; Micek, P.; Morin, P.; Ueckerdt, T.; Wood, D. R.: Planar graphs have bounded queue-number. Journal of the ACM, 67, 1–38 (2020) 8. Dujmovic, V., Wood, D. R.: On linear layouts of graphs, Discrete Mathematics and Theoretical Computer Science, 6 (2), 339–357 (2004) 9. Harary, F.: Graph Theory, Addison Wesley, Reading, Mass. (1969) 10. Heath, L. S., Rosenberg, A. L.: Laying out graphs using queues. SIAM J. Comput., 21 (5), 927–958 (1992) 11. Ollmann, L. T.: On the book thicknesses of various graphs, Proceedings of the 4th Southeastern Conference on Combinatorics, Graph Theory and Computing, 8, 459 (1973) 12. Overbay, S.: Embedding graphs in cylinder and torus books, Journal of Combinatorial Mathematics and Combinatorial Computing, 91, 299–313 (2014) 13. Overbay, S.: Generalized book embeddings, Ph.D. Dissertation, Colorado State University, Fort Collins, CO, (1998) 14. Pupyrev, S.: Book embeddings of graph products, arXiv:2007.15102 (2020) 15. Reghizzi, S. C., Pierluigi, S. P.: Deque automata, languages, and planar graph representations, Theoretical Computer Science, 834, 43–59 (2020) 16. Yang, J., Shao, Z.; Li, Z.: Embedding Cartesian products of some graphs in books, Commun. Math. Res., 34, 253–260 (2018)

Möbius Book Embeddings Nicholas Linthacum, Luke Martin, Thomas McKenzie, Shannon Overbay, and Lin Ai Tan

Abstract Book embeddings of graphs have been the subject of extensive study. In this chapter we generalize the definition of such embeddings by allowing book pages to be Möbius strips, rather than half planes. We give an application of these Möbius books and we give page bounds for complete graphs and bipartite graphs. We conclude with optimal Möbius book embeddings of some well-known graph families. Keywords Graph theory · Book thickness · Möbius book embedding Mathematics Subject Classification (2000) 05C10, 05C90, 68R10

1 Background and Definitions The concept of a book embedding of a graph was introduced in [3] and [5]. To construct a book embedding, the vertices of a graph are ordered in a line which represents the spine of the book. A page in the book is a half plane on which edges of the graph are placed so that no two edges cross each other or the spine. The book thickness of a graph is the smallest number of pages needed to embed the graph in this manner. We write .bt (G) for the book thickness of a graph G. Overbay considered book embeddings where the pages of the book were cylinders or tori rather than half-planes in [6]. In [2] the authors showed that a one page cylinder embedding can be thought of as a data structure called a deque, while

N. Linthacum · T. McKenzie · S. Overbay (✉) · L. A. Tan Gonzaga University, Spokane, WA, USA e-mail: [email protected]; [email protected]; [email protected]; [email protected] L. Martin University of Kentucky, Lexington, KY, USA e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 S. Heuss et al. (eds.), Combinatorics, Graph Theory and Computing, Springer Proceedings in Mathematics & Statistics 462, https://doi.org/10.1007/978-3-031-62166-6_26

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in [4], the authors showed that a one page torus embedding can be thought of as a data structure they called a torus deque. This chapter will examine book embeddings in which the pages are Möbius bands and will show that a one page Möbius book can be thought of as a data structure we will call a Möbius deque.

2 Möbius Books Throughout this chapter we write .Kn for the complete graph on n vertices and .Km,n for the complete bipartite graph with vertex sets of sizes m and n. To construct a one page embedding of a graph, we draw a line called the spine across the width of a Möbius band and then we place the vertices of the graph on the line. If we cut the band along the line, we can represent this construction with the picture in Fig. 1. Next we place the edges of the graph on the band in such a way that no edge crosses the spine and no two edges cross each other. If this is not possible, we try again using a different ordering of the vertices. If no ordering of the vertices yields an embedding without crossings, then the graph is not embeddable in a one page Möbius book. Figure 2 gives an example of a one page Möbius book embedding of a graph. Note that if there are no edges from top to bottom, this is equivalent to a (traditional) two page book, which is a planar structure. Placing edges from top to bottom allows us to embed certain non-planar graphs as well. Fig. 1 A Möbius page

Fig. 2 A one page Möbius book embedding of a graph

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An n page Möbius book is a collection of n one page Möbius books such that the vertex ordering is identical on each page. In a Möbius book embedding of a graph, each edge is assigned to exactly one page of the Möbius book. The minimum number of Möbius pages needed to embed a graph G, taken over all vertex orderings and edge assignments, is called the Möbius book thickness of G and is written .mbt (G). This definition is a natural extension of previously defined books (see [2– 5], and [6]). Given a graph G we write .bt (G) for its book thickness, .cbt (G) for its cylinder book thickness, and .tbt (G) for its torus book thickness.

3 An Application Figure 1 suggests an application of Möbius books. If we look at the left half of this figure, we can think of the vertices as representing offices arranged across from each other in a hallway, as shown in Fig. 3. Note that we may assume the number of offices, n, is even, since we may always add an extra zero degree vertex. So office .v2 is directly across from office .vn−1 , while offices .v1 and .v3 are adjacent to .v2 . Call one side of the hallway the north side (N) and the other the south (S). Now imagine a cart with a red side (R) and a blue side (B) moving down the hall from left to right. Say the red side faces north and the blue side faces south. Offices .vn , . . . , v(n/2)+1 are lined up on the north side and offices .v1 , . . . , vn/2 are lined up on the south. As the cart arrives at an office, the occupant can slide an object onto the side of the cart facing them, take an object off the side of the cart facing them, or do nothing. This is subject to the constraint that one object on the cart can not be slipped past another. When the cart gets to the end of the hallway, it reverses orientation, so that red now faces south and blue north. The cart now makes a return journey, from right to left. Once the cart returns to its starting position, it should be empty. Table 1 shows how this works with the embedding from Fig. 2. Figure 4 shows what this situation looks like at step 6, right after office B puts .e6 on the cart. If there is ambiguity, we use the conventions given in [2] to determine in what order we remove and add edges.

Fig. 3 Möbius book application

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Table 1 Application of Fig. 2 Application of Fig. 2 Step Cart NR SB 0 1 NR .e1 , e2 , e3 SB 2 NR .e1 , e2 , e3 SB NR .e5 , e4 SB 3 4 5 6 7

NB .e4 , e5 SR NB .e5 SR NB .e5 , e6 SR NB SR

Action Starting position Office F puts .e1 on. Office A puts .e2 on and .e3 on No additions or removals Office D removes .e1 and then .e2 . Office C removes .e3 , and then adds .e5 followed by .e4 Cart turns around Office D pulls .e4 off Office B puts .e6 on Office F pulls .e5 off and .e6 off

Fig. 4 Step 6

4 An Edge Bound Edge bounds for one page cylinder and torus books can be found in [6]. Given n vertices, at most .3n − 6 edges can be placed on one cylinder page and at most 3n edges can be placed on a torus page. We begin this section by giving an edge bound for a one page Möbius book. Theorem 1 If a graph G has n vertices, then at most .3n − 3 edges of G can be placed on one Möbius page. Proof Figure 5 shows that we can always place n edges along the top and one side of Fig. 1 without obstructing any other edge. We can draw duplicate copies of these edges on the bottom and the other side of Fig. 1 as well, forming a boundary cycle as shown in Fig. 6. All remaining edges must be placed within this bounding cycle of length 2n. The maximum number of edges would be given by a triangulation of the interior of this cycle, which would add at most .2n − 3 distinct edges. Thus there are at most .n + (2n − 3) = 3n − 3 possible edges on a single Möbius page. ⨆ ⨅ We note that this edge bound is the same as the one that may be derived from Euler’s formula on a Möbius strip (which is a punctured projective plane). The difference here is that we restrict the vertices to a line. Even with this restriction

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Fig. 5 Border edges

Fig. 6 Bounding rectangle

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of the vertex placement, we show that this bound is sharp. The next theorem shows that .K6 , which has 6 vertices and .3 · 6 − 3 edges, attains the edge bound and admits a one page Möbius book embedding. Theorem 2 The complete graph on six vertices, .K6 , has Möbius book thickness one. Proof Figure 7 gives an embedding.

⨆ ⨅

So .K6 , which is not a planar graph, is the largest complete graph embeddable on a one page Möbius book. Extending Theorem 1 to a k page Möbius book, we obtain the following edge bound. Theorem 3 If a graph G has n vertices, then at most .n + k(2n − 3) edges of G can be placed on a k page Möbius book. Proof We can embed at most .2n − 3 edges, corresponding to a triangulation, on each of the k pages. As in the proof of Theorem 1, we can embed the n border edges on any page. ⨆ ⨅

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Solving for k in the previous theorem yields the following result. Corollary 1 If a graph G has n vertices and e edges, then the Möbius book e−n . thickness of G, .mbt (G), is greater than or equal to the ceiling of . 2n−3 This gives the following bound for the Möbius book thickness of the complete graph. Corollary 2 If .n > 3, ⎾ .

n2 − 3n 4n − 6

⏋ ≤ mbt (Kn ) ≤

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edges. Plugging this into the last corollary yields the Proof .Kn has .e = n(n−1) 2 inequality on the left. By [3], .bt (Kn ) = ⎾ n2 ⏋, when .n > 3. Observing that each Möbius page contains two standard book pages yields the inequality on the right. ⨆ ⨅ This last result implies that the Möbius book thickness is growing on the order of n/4. When .n = 10, 11, and 12 the corollary gives an exact Möbius book thickness of 3. Whereas when .n = 9, the corollary tells us .2 ≤ mbt (K9 ) ≤ 3. In fact, .mbt (K9 ) = 2 as shown in Fig. 8. At this point is natural to consider bipartite graphs. We apply Euler’s formula to bipartite graphs to get the following well-known theorem. .

Theorem 4 If a bipartite graph with n vertices and e edges can be embedded on a Möbius strip, then .2n − 2 ≥ e. Fig. 8 .K9

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Fig. 9 .K3,4

Note that the previous theorem is a statement about arbitrary embeddings of bipartite graphs in a Möbius strip. Nonetheless it provides an upper bound for embeddings in a one page Möbius book. Corollary 3 If a bipartite graph with n vertices and e edges can be embedded in a one page Möbius book, then .2n − 2 ≥ e. Embeddings in a projective plane were studied extensively by Archdeacon in [1] who classified all forbidden minors for such embeddings. These forbidden minors include .K3,5 and .K4,4 which violate this edge bound. The graph .K3,4 has 12 edges and 7 vertices, so it satisfies .e = 2n − 2. The next result gives a one page Möbius book embedding. Theorem 5 The complete bipartite graph .K3,4 has Möbius book thickness one. Proof Figure 9 gives an embedding.

⨆ ⨅

Recalling that any graph embeddable in a standard two page book can be embedded in a one page Möbius book and observing that the complete bipartite graphs .K1,n and .K2,n are two page embeddable, we now have the following theorem. Theorem 6 The complete bipartite graph .Km,n has Möbius book thickness one if and only if it is isomorphic to .K1,n , .K2,n , .K3,3 , or .K3,4 .

5 Other Graphs With Möbius Book Thickness One We conclude this chapter by listing a few graphs which are not embeddable in a (traditional) two page book, but are embeddable on a one page Möbius book. Theorem 7 The second stellation of the triangle .St 2 (K3 ) (the Goldner-Harary graph), has Möbius book thickness one. Proof Figure 10 gives a one page Möbius embedding. See [3] for a proof that this graph has (traditional) book thickness 3. ⨆ ⨅

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Fig. 10 .St 2 (K3 )

Fig. 11 Möbius ladder

Fig. 12 One page embedding of Möbius ladder

The Möbius ladder is defined for an even number of vertices as shown in Fig. 11. These graphs are non-planar when the number of vertices is greater than four. Theorem 8 All Möbius ladders have Möbius book thickness one. Proof See Fig. 12.

⨆ ⨅

Theorem 9 All circulant graphs of the form .C(n, {1, 2}) have Möbius book thickness one. Proof When n is even, .n = 2k, the interior of the graph is composed of two disjoint cycles, one connecting the odd-labeled vertices and one connecting the even-labeled vertices. One of these cycles can be drawn on the outside of the graph to create a

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Fig. 13 .C(2k + 1, {1, 2})

U2k+1

U1

U2k

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U3

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Fig. 14 One page Möbius book embedding of .C(2k + 1, {1, 2})

Fig. 15 .P (2n, n)

planar structure. Then, .C(2k, {1, 2}) is a planar graph with a Hamiltonian cycle, meaning it can be embedded in a 2-page book (see [3]). Since a Möbius book contains a 2-page book, it can be embedded in one Möbius page. When n is odd, .n = 2k + 1 and the standard drawing of the graph is given in Fig. 13. In this case, the graph is non-planar, but admits a one page Möbius embedding as shown in Fig. 14. ⨆ ⨅ Theorem 10 All Petersen graphs of the form .P (2n, n) (Fig. 15) have Möbius book thickness one.

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Fig. 16 One page Möbius book embedding of .P (2n, n) Fig. 17 .P (2k + 1, 2)

Fig. 18 One page Möbius book embedding of .P (2k + 1, 2)

Proof Figure 16 gives an embedding. To find this embedding, we used the fact that .P (2n, n) is homeomorphic to .C(2n, {1, n}), which is isomorphic to the Möbius ladder. The one page embedding of .P (2n, n) is an extension of our Möbius ladder’s one page embedding. ⨆ ⨅ Theorem 11 All Petersen graphs of the form .P (n, 2) have Möbius book thickness one. Proof When n is even, .n = 2k, the inside vertices are connected by two disjoint cycles. Thus, as with .C(2k, {1, 2}), we can embed .P (n, 2) in a one page Möbius ⨆ ⨅ book. When n is odd, Figs. 17 and 18 give an embedding of .P (2k + 1, 2).

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6 Future Work Regular book embeddings are the most extensively studied book embeddings of graphs. Additional work has been done with cylinder and torus book embeddings, and we have extended the family of books to Möbius books in this chapter. In the future, we hope to study Klein book embeddings by placing the spine along the handle of a Klein bottle or by wrapping the spine around the handle. There could be interesting similarities and differences between books with orientable vs. non-orientable pages. Furthermore, we anticipate an application of Klein books to delivery systems analogous to the application given in Sect. 3. Acknowledgments Student funding for this project was provided through the Gonzaga University McDonald Work Award.

References 1. Archdeacon, D.: A Kuratowski theorem for the projective plane. The Ohio State University, 1980. 2. Auer, C.; Bachmaier, C.; Brandenburg, F.J.; Brunner W.; Gleißner, A.: Data structures and their planar layouts, Journal of Graph Algorithms and Applications, 22 no. 2 (2018), 207–237. https://doi.org/10.7155/jgaa.00465 3. Bernhart, F.; Kainen, P.: The book thickness of a graph, J. Combin. Theory Ser. B 27 (1979), 320–331. https://doi.org/10.1016/0095-8956(79)90021-2 4. McKenzie, T.; Overbay, S.: Dequeues on a Torus, in Combinatorics, Graph Theory, and Computing, Chapter 25. Springer, Cham, 2024. 5. Ollmann, L.: On the book thicknesses of various graphs, Proc. 4th Southeastern Conf. on Combinatorics, Graph Theory and Comput. 8 (1973), 459. 6. Overbay, S.: Embedding graphs in cylinder and torus books, J. Combin. Math. Combin. Comput., 91 (2014), 299–313.

DNA Self-assembly: Friendship Graphs Leyda Almodóvar, Emily Brady, Michaela Fitzgerald, Hsin-Hao Su, and Heiko Todt

Abstract Based on the flexible tile method for DNA self-assembly, we find a collection of tiles that will construct a nanostructure shaped like a friendship graph or a dumbbell-friendship graph. We find the minimum number of tile and bond-edge types required to construct these graphs in three different scenarios representing distinct levels of laboratory constraints. Keywords DNA self-assembly · Branched junction molecules · Friendship graphs

1 Introduction DNA self-assembly is the process of spontaneous formation of nanostructures from carefully designed branched DNA molecules. DNA self-assembly, and selfassembly in general, is a rapidly advancing field with [1, 2] providing good overviews. While there are several methods for the construction of self-assembled nanostructures [3, 4], we focus on the flexible-tile model introduced in [5]. In this chapter, we focus on the graph-theoretical aspect of designing optimal tiles that will construct friendship graphs and dumbbell-friendship graphs in different scenarios. The tiles are branched junction molecules whose flexible k-arms are double strands of DNA, represented abstractly as a vertex of degree k in a graph. These arms have cohesive-ends which can bond to any other cohesive-end with a complementary sequence of bases. The cohesive-ends are represented with letters, called bond-edge types, with complementary Watson-Crick bases represented by hatted and unhatted letters, respectively. For example, given the bond-edge type a, its complementary sequence of bases is represented by .a. ˆ A collection of tiles, called a pot, realizes a graph G, if the collection can be assembled to construct

L. Almodóvar (✉) · E. Brady · M. Fitzgerald · H.-H. Su · H. Todt Stonehill College, North Easton, MA, USA e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 S. Heuss et al. (eds.), Combinatorics, Graph Theory and Computing, Springer Proceedings in Mathematics & Statistics 462, https://doi.org/10.1007/978-3-031-62166-6_27

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G without any arms remaining unmatched. The tile type is the multiset of letters corresponding to bond-edge types for the tile. Our goal is to find the minimum number of tile and bond-edge types needed in order to construct a target graph G, where G is a generalized friendship graph or a dumbbell-friendship graph. In [6] and [7], the authors explored these questions for triangular lattices. The problem became extremely complicated as the triangle lattices became larger. We explore whether rearranging the connection between triangles in lieu of obtaining lattices or considering other shapes like squares will allow us to find a general pattern for the pots. We consider these minimum numbers under three different scenarios corresponding to three laboratory constraints: – Scenario 1: A graph with a fewer number of vertices than the target graph may be realized from a pot of tiles. – Scenario 2: A graph with the same or greater number of vertices, but not isomorphic to the target graph may be realized from a given pot of tiles. No graphs with fewer vertices than the target graph may be realized. – Scenario 3: No graph with fewer vertices nor non-isomorphic graphs with the same number of vertices as the target graph may be realized from a given pot of tiles. While we focus on the theoretical aspect of designing tiles, it should be noted that in a laboratory setting, the first two scenarios allow by-product DNA nanostructures of smaller or equal size to the selected branched junction molecule, but these need not be shaped like the target graph. This is not possible in scenario 3. We define .Ti (Gn ) for .i = 1, 2, 3 as the minimum number of tiles required to construct a complex in each of the scenarios above. Similarly, .Bi (Gn ) denotes the minimum number of bond-edge types needed for each scenario. We show how different scenarios differ in Example 1. Example 1 Consider the graph G shown in Fig. 1. The pot .P = {{a, a}, ˆ {aˆ 2 , a 3 }, {aˆ 3 , a 2 }} realizes G in scenario 1 as can be seen in Fig. 2. But, this pot does not satisfy scenarios 2 or 3 as the tile .{a, a} ˆ can also ˆ It realize the graph with a single vertex and a loop by connecting the ends a and .a. can be shown that .B1 (G) = 1 and .T1 (G) = 3. ˆ {a, ˆ realizes G as can be seen in Fig. 3. It The pot .P = {{a 4 , b}, {a 4 , b}, ˆ b}, {a, ˆ b}} is not possible to realize a graph with less than ten vertices with this pot. However, it is possible to construct the graph in Fig. 4, which is not isomorphic to G. Therefore, Fig. 1 Graph G

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Fig. 2 A pot realizing G in scenario 1

Fig. 3 A pot realizing G in scenario 2

Fig. 4 Graph non-isomorphic to G

Fig. 5 A pot realizing G in Scenario 3

this pot satisfies scenario 2, but not scenario 3. With some additional work, it can be shown that .B2 (G) = 2 and .T2 (G) = 4. ˆ {d, ˆ b}} ˆ realizes G, as Finally, the pot .P = {{a, ˆ c}, {a, ˆ c}, ˆ {a 5 }, {a, ˆ b4 }, {d, b}, shown in Fig. 5. It can be verified exhaustively that any graph realized by this pot has at least ten vertices and if it has ten vertices, then it is isomorphic to G. It can be shown that .B3 (G) = 4 and .T3 (G) = 6, and we will do so in Sect. 6.2.

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2 Definitions and Prior Results In this section we include definitions and results from [8] that will be used throughout our chapter. We have divided these according to the scenarios where they are needed.

2.1 Scenario 1 Corollary 1 B1 (G) = 1 for all G. Theorem 1 av(G) ≤ T1 (G) ≤ ev(G) + 2ov(G), where av(G), ev(G), and ov(G) are the number of different degrees, different even degrees, and different odd degrees of the graph G, respectively.

2.2 Scenario 2 Definition 1 Let P be a pot with p tile types labeled t1 , . . . , tp , let Aij be the number of cohesive ends of type ai on tile tj , let Aˆ i,j be the number of cohesive ends of type aˆ i , and let zi,j = Ai,j − Aˆ i,j . The construction matrix of P , denoted M(P ) is given by ⎡

z1,1 z1,2 . . . z1,p ⎢ .. .. .. ⎢ . . . .M(P ) = ⎢ ⎣zm,1 zm,2 . . . zm,p 1 1 ... 1

0

⎤

⎥ ⎥ ⎥ 0⎦ 1

and it captures the requirements of a complete complex. That is, the sum of the proportion of tile types needs to add up to 1, and the total number of hatted cohesive end types must equal the total number of unhatted cohesive-end types. Proposition 1 Let P be a pot with p tile types labeled t1 , . . . , tp . Let ri to be the proportion of tile type ti used in the assembly process. If 〈r1 , . . . , rp 〉 is a solution of the construction matrix M(P ), and there is a positive integer n such that nrj ∈ Z≥0 for all j , then there is a graph of size n that may be constructed from P using nrj tiles of type tj . The construction matrix of a given pot P along with Proposition 1 allows us to determine the smallest sized graph that may be constructed from P . This is particularly useful in scenario 2 since the realization of smaller graphs than a target graph G are not allowed. Theorem 2 B2 (G) + 1 ≤ T2 (G).

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2.3 Scenario 3 Definition 2 Given a pot P , the set of graphs of minimum size that may be constructed from P is denoted Cmin (P ). Lemma 1 If P is a pot such that {G} = Cmin (P ) and G has no loops, then no tile type T ∈ P used in the construction of G may be used for two adjacent vertices in G. Lemma 2 If P is a pot such that {G} = Cmin (P ), and two nonadjacent edges {u, v} and {s, t} of G = {V , E} use the same bond-edge type, then G is isomorphic to G' = {V , E ' }, where E ' = E − {{u, v}, {s, t}} ∪ {{u, t}, {s, v}}. The following theorem gives us a relation between the minimum number of tiles and bond-edge types needed respectively in each scenario: Theorem 3 B1 (G) ≤ B2 (G) ≤ B3 (G) and T1 (G) ≤ T2 (G) ≤ T3 (G).

2.4 Friendship and Generalized Friendship Graphs In this chapter, we investigate the three scenarios for friendship graphs and other related graphs. A friendship graph .Fn consists of n copies of cycle graphs .C3 joined at a common vertex. A friendship graph is also known as a windmill graph. Friendship graphs were first discussed in [9], where it was shown that any graph where any two vertices have exactly one adjacent vertex (“a friend”) in common must be friendship graphs. A generalized friendship graph can be constructed by joining n copies of .Cm with a common vertex, denoted as .GFm,n where m denotes the number of edges in each cycle and n denotes the number of cycles. Note that .GF3,n = Fn . Figure 6 shows the graphs of two generalized friendship graphs.

Fig. 6 The generalized friendship graphs .GF4,3 and .GF3,4

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3 Friendship Graphs In this section, we consider the different scenario for friendship graphs .Fn where n > 1. If .n = 1 then .F1 = C3 , and results for cycles are already known and can be found in [8].

.

3.1 Scenario 1 Consider the graph .Fn with .n > 1, shown in Fig. 7. Note that .Fn has 2n vertices of degree two and one vertex of degree 2n. Therefore .T1 (Fn ) ≥ 2. The pot .P = {{a, a}, ˆ {a n , aˆ n }} realizes .Fn . Hence, .T1 (Fn ) = 2.

3.2 Scenario 2 ˆ can The pot from scenario one does not satisfy scenario two since the tile .{a, a} construct a graph with one vertex and a loop by connecting a and .aˆ as shown in Fig. 8. If we label one of the degree two vertices as .{a 2 } and the other as .{aˆ 2 }, then we will be able to construct a graph of size 2. Thus, .B2 (Fn ) > 1. This shows that .B2 (Fn ) ≥ 2. By Theorem 2, .T2 (Fn ) ≥ 3. ˆ realizes .Fn as shown in Fig. 9. While it is The pot .P = {{a 2n }, {a, ˆ b}, {a, ˆ b}} clear in this case that the pot does not realize a graph smaller than .Fn , we can prove this statement with the construction matrix, shown here to illustrate this technique. ⎡ ⎤ 2n −1 −1 0 .M(P ) = ⎣ 0 1 −1 0⎦ 1 1

1 1

Fig. 7 A graph realizing .Fn in scenario 1

Fig. 8 The incidental construction of a graph using the tile .{a, a} ˆ

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Fig. 9 A pot realizing .Fn in scenarios 2 and 3

⎡ ⎤ 1 1 0 0 2n+1 ⎢ n ⎥ .rref(M(P )) = ⎣0 1 0 2n+1 ⎦ n 0 0 1 2n+1 n 1 , and .r2 = 2n+1 = r3 . Using From this matrix, we can see that .r1 = 2n+1 Proposition 1, we conclude that the smallest graph that can be constructed from the pot is of size .2n + 1. Hence, .B2 (Fn ) = 2. By Theorem 2, .T2 (Fn ) = 3.

3.3 Scenario 3 By Theorem 3, we know that .T3 (Fn ) ≥ 3 and .B3 (Fn ) ≥ 2. The pot .P = ˆ realizes .Fn in scenario 3 since no non-isomorphic graphs {{a 2n }, {a, ˆ b}, {a, ˆ b}} can be constructed from this pot. Hence, we can conclude that .T3 (Fn ) = 3 and .B3 (Fn ) = 2.

4 Generalized Friendship Graphs GF4,n In this section, we investigate generalized friendship graphs where the cycles are of length 4.

4.1 Scenario 1 Consider the graph .GF4,n with .n > 1, shown in Fig. 10. Note that .GF4,n has 3n vertices of degree two and one vertex of degree 2n. Therefore .T1 (GF4,n ) ≥ 2. The pot .P = {{a, a}, ˆ {a n , aˆ n }} realizes .GF4,n . Hence, .T1 (GF4,n ) = 2.

4.2 Scenario 2 By a similar argument from Sect. 3.2, .B2 (GF4,n ) > 1.

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Fig. 10 The labeling of in scenario 1

.GF4,n

Fig. 11 A pot realizing in scenarios 2 and 3

.GF4,n

The pot .P = {{a 2n }, {a, ˆ b}, {bˆ 2 }} realizes .GF4,n as shown in Fig. 11. We can prove this using the construction matrix. ⎡ ⎤ 2n −1 0 0 .M(P ) = ⎣ 0 1 −2 0⎦ 1 1 1 1 ⎤ ⎡ 1 1 0 0 3n+1 ⎢ 2n ⎥ .rref(M(P )) = ⎣0 1 0 3n+1 ⎦ n 0 0 1 3n+1 1 2n n From this matrix we can see that .r1 = 3n+1 , .r2 = 3n+1 , and .r3 = 3n+1 . Using Proposition 1, we conclude that the smallest graph that can be constructed from the pot is of size .3n + 1. Hence, .B2 (GF4,n ) = 2. By Theorem 2, .T2 (GF4,n ) = 3.

4.3 Scenario 3 By Theorem 3, we know that .T3 (GF4,n ) ≥ 3 and .B3 (GF4,n ) ≥ 2. The pot P = {{a 2n }, {a, ˆ b}, {bˆ 2 }} also realizes .GF4,n in scenario 3 since no non-isomorphic graphs can be constructed from this pot by symmetry. Hence, we can conclude that .T3 (GF4,n ) = 3 and .B3 (GF4,n ) = 2. .

5 Generalized Friendship Graphs GFm,n for m ≥ 5 We present results for .Ti and .Bi for generalized friendship graphs .GFm,n for .m ≥ 5.

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Fig. 12 A pot realizing in scenarios 2 and 3

.GF5,n

Proposition 2 .T1 (GFm,n ) = 2. Proof Consider the graph .GFm,n with .n > 1. Note that .GFm,n has .(m−1)n vertices of degree two and one vertex of degree 2n. Therefore .T1 (GFm,n ) ≥ 2. The pot .P = {{a, a}, ˆ {a n , aˆ n }} realizes .GFm,n . Hence, .T1 (GFm,n ) = 2. Proposition 3 .B2 (GFm,n ) = ⎾ n2 ⏋ and .T2 (GFm,n ) = ⎾ n2 ⏋ + 1. Proof Note that to maximize the use of bond-edge types, our approach is to label all the .Cm cycles identically so that any swapping of edges results in a graph of the same size. Suppose .B2 (GFm,n ) < ⎾ n2 ⏋. Then, at least one bond-edge type appears at least three times in the complete complex forming a copy of .Cm . At least two of the three edges must have the same orientation going around the cycle. These two edges may detach and rejoin to make two smaller-size complexes. Therefore, .B2 (GFm,n ) = ⎾ n2 ⏋. The pot .Peven = {{a12n , {aˆ 1 , a2 }, {aˆ 2 , a3 }, . . . {aˆ n−1 , aˆ n+1 }} realizes .GFm,n for 2

2

m ≥ 5 odd and the pot .Podd = {{a12n , {aˆ 1 , a2 }, {aˆ 2 , a3 }, . . . {aˆ n2 , aˆ n2 }} realizes .GFm,n for .m ≥ 6. An example for .GF5,n is shown in Fig. 12. It can be verified that the construction matrix that the smallest graph this pot can create has size .(m−1)n+1. By Proposition 1, .B2 (GFm,n ) = ⎾ n2 ⏋ and .T2 (GFm,n ) = ⎾ n2 ⏋ + 1.

.

Proposition 4 .B3 (GFm,n ) = ⎾ n2 ⏋ and .T3 (GFm,n ) = ⎾ n2 ⏋ + 1. Proof We know .B3 (GFm,n ) ≥ ⎾ n2 ⏋ and .T3 (GFm,n ) ≥ ⎾ n2 ⏋ + 1 by Theorem 3 and Proposition 3. If P is one of the pots presented in Proposition 3, then .{GFm,n } = Cmin (P ). Hence, .B3 (GFm,n ) = ⎾ n2 ⏋ and .T3 (GFm,n ) = ⎾ n2 ⏋ + 1.

6 Dumbbell-Friendship Graphs with n C3 Cycles on Each Side After finding results for generalized friendship graphs, we focus on similar graphs to see if we can find .Ti and .Bi for graphs containing more copies of .Cm , where the .Cm ’s are distributed a bit differently. The dumbbell graph is the union of two .Cm cycles connected by an additional edge between one vertex from each cycle. From here, we can construct the dumbbell-friendship graph, which is the union of one friendship graph with another friendship graph by an extra edge that connects the two common vertices. We denote a dumbbell-friendship graph with n .Cm cycles as .DFm,n .

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Fig. 13 A pot realizing in scenario 1

.DF3,n

In this section, we present results for dumbbell-friendship graphs .DF3,n and in the subsequent section we consider .DF4,n .

6.1 Scenario 1 Consider the dumbbell-friendship graph .DF3,n . Note that .DF3,n has 4n vertices of degree two and two vertices of degree .2n + 1. Therefore .2 ≤ T1 (DF3,n ) ≤ 3 by Theorem 1. Suppose .T1 (G) = 2. This implies that both tiles will be reused and the degree two tile must be .{a, a} ˆ since it connects to itself. But, this forces two distinct degree .n + 1 tile types since there is an odd number of edges. Hence, .T1 (G) = 3. ˆ {a n+1 , aˆ n }{a n , aˆ n+1 }} realizes .DF3,n (Fig. 13). The pot, .P = {{a, a},

6.2 Scenario 2 Note that .B2 (DF3,n ) > 1 since a graph of size one might be created from the tile {a, a} ˆ and a graph of size two might get created from the tiles .{a 2 } and .{aˆ 2 }.The ˆ {a, ˆ realizes .DF3,n in scenario 2, as shown in pot .P = {{a n , b}, {a n , b}, ˆ b}, {a, ˆ b}} Fig. 14. We can prove this using the construction matrix. The construction matrix of P is shown below. ⎡ ⎤ n n −1 −1 0 .M(P ) = ⎣ 1 −1 1 −1 0⎦ 1 1 1 1 1 ⎤ ⎡ 1−n 1 0 0 −1 2n+2 ⎢ 1 ⎥ .rref(M(P )) = ⎣0 1 0 1 2 ⎦ n 0 0 1 1 n+1

.

This matrix yields the system of equations shown below, where .r4 is a free variable represented as s. ⎧ 1−n ⎪ r1 = 2n+2 +s ⎪ ⎪ ⎪ ⎨r = 1 − s 2 2 . n ⎪ = r ⎪ 3 n+1 − s ⎪ ⎪ ⎩ r4 = s

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Fig. 14 A pot realizing in scenario 2

.DF3,n

Fig. 15 Graph non-isomorphic to .DF3,n

1 1 n n Using software presented in [7], we found that .〈 2n+2 , 2n+2 , 2n+2 , 2n+2 〉 is a n solution of .M(P ) and .s = 2n+2 is the smallest possible value of s. By Proposition 1, we conclude that the smallest graph that can be constructed from the pot is of size .2n + 2. Hence, .B2 (DF3,n ) = 2. Since .T1 (DF3,n ) = 3, we know by Theorem 3 that .3 ≤ T2 (DF3,n ) ≤ 4. If .T2 (DF3,n ) = 3, then there must be two distinct tile types of degree .n + 1 and only one tile type of degree two by the argument presented in scenario 1. But, the degree two tile connects to itself, which implies that it must be .{a, a}, ˆ which is not allowed in scenario 2. Hence, .T2 (DF3,n ) = 4.

6.3 Scenario 3 By Lemma 1, .T3 (DF3,n ) ≥ 4. If a degree two tile is used more than once on opposite sides of the middle edge connecting the two friendship graphs, we would be able to construct a non-isomorphic graph of the same size by “breaking” our original edges and reconnecting them in a different manner as seen in Fig. 15. To prevent this from happening, we will need a distinct degree two tile for friendship graphs on opposite sides of the middle edge connecting the friendship graphs. In order to minimize the number of bond-edge types used, we can start by labeling the two degree .n + 1 tiles. To maximize the amount of times we reuse the same bond-edge type, we can label them as .{a n+1 } and .{a, ˆ bn }, respectively. The labels a and b cannot be used for the degree two vertices by Lemma 2. Thus, we must introduce a new bond-edge type c for the degree two tiles. Following this process, we obtain n+1 }, {a, ˆ d}, {b, ˆ d}}. ˆ Using the construction matrix, it .P = {{a ˆ bn }, {a, ˆ c}, {a, ˆ c}, ˆ {b, can be shown that the smallest graph that can be created from this pot is of size .2n +

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Fig. 16 A pot realizing in scenario 3

.DF3,n

2. Additionally, it can be verified by a straightforward combinatorial argument that no non-isomorphic graphs may be created. Figure 16 shows how this pot realizes .DF3,n . Therefore, .T3 (DF3,n ) = 6 and .B3 (DF3,n ) = 4.

7 Dumbbell-Friendship Graphs with n C4 Cycles on Each Side 7.1 Scenario 1 Consider the dumbbell-friendship graph .DF4,n . Notice it has two vertices of degree 2n + 1 and 6n vertices of degree two. Therefore .2 ≤ T1 (DF3,n ) ≤ 3 by Theorem 1. Suppose .T1 (G) = 2. This implies that both tiles will be reused and the degree two ˆ since it connects to itself. But, this forces two distinct degree tile must be .{a, a} .n + 1 tile types since there is an odd number of edges. Hence, .T1 (G) = 3. The pot, .P = {{a, a}, ˆ {a n+1 , aˆ n }{a n , aˆ n+1 }} realizes .DF4,n , as shown in Fig. 17. .

7.2 Scenario 2 Note that .B2 (DF4,n ) > 1 since a graph of size one might be created from the tile {a, a} ˆ and a graph of size two might get created from the tiles .{a 2 } and .{aˆ 2 }.The pot n n ˆ {a, .P = {{a , b}, {a , b}, ˆ b}, {bˆ 2 }} realizes .DF4,n in scenario 2, as shown in Fig. 18. We can prove this using the construction matrix by a similar argument shown in Sect. 6.2. We conclude that the smallest graph that can be constructed from the pot is of size .3n + 2. Hence, .B2 (DF4,n ) = 2. Since .T1 (DF4,n ) = 3, we know by Theorem 3 that .3 ≤ T2 (DF4,n ) ≤ 4. If .T2 (DF4,n ) = 3, then there must be two distinct tile types of degree .n + 1 and only one tile type of degree two by the argument presented in scenario 1. But, the degree two tile connects to itself, which implies that it must be .{a, a}, ˆ which is not allowed in scenario 2. Hence, .T2 (DF4,n ) = 4. .

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Fig. 17 A pot realizing in scenario 1

.DF4,n

Fig. 18 A pot realizing in scenario 2

.DF4,n

Fig. 19 A pot realizing in scenario 3

.DF4,n

7.3 Scenario 3 By Theorem 3, .T3 (DF4,n ) ≥ 4. By a similar argument from Sect. 6.3, it can be ˆ d}, {dˆ2 }} realizes .DF4,n shown that the pot .P = {{a n+1 }, {a, ˆ bn }, {a, ˆ c}, {cˆ2 }, {b, and no smaller or non-isomorphic graph may be created from this pot. Therefore, .T3 (DF4,n ) = 6 and .B3 (DF4,n ) = 4 (Fig. 19).

8 Conclusion We obtained complete results for all scenarios for .Fn , and .GFm,n for .m ≥ 4 and for the dumbell-friendship graphs .DF3,n , and .DF4,n . A similar approach used to find .Ti (GFm,n ) and .Bi (GFm,n ) should provide upper bounds for .Ti (DFm,n ) and .Bi (DFm,n ) for .i = 1, 2, 3.

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Acknowledgments We would like to thank the Stonehill Undergraduate Research Experience program (SURE) for support and funding to make this chapter possible.

References 1. Pelesko, John A: Self assembly: the science of things that put themselves together. Chapman and Hall/CRC (2007) 2. Seeman, Nadrian C: An overview of structural DNA nanotechnology. Molecular biotechnology vol. 37(3), pp. 246. Springer (2007) 3. Chen, Junghuei and Seeman, Nadrian C: Synthesis from DNA of a molecule with the connectivity of a cube. Nature, vol. 350(6319) pp. 631–633. Springer (1991) 4. Jonoska, Natasa and Karl, Stephen A and Saito, Masahico: Three dimensional DNA structures in computing. BioSystems, vol. 52, pp. 143–153. Elsevier (1999) 5. Jonoska, Natasa and McColm, Gregory L and Staninska, Ana: Spectrum of a pot for DNA complexes. International Workshop on DNA-Based Computers, pp. 83–94. Springer (2006) 6. Almodóvar, Leyda and Mauro, Samantha and Martin, Sydney and Todt, Heiko: Minimal tile and bond-edge types for self-assembling DNA graphs of triangular lattice graphs. Congressus Numeratium vol. 232, pp. 241–263. (2019) 7. Almodóvar, Leyda and Ellis-Monaghan, Jo and Harsy, Amanda and Johnson, Cory and Sorrells, Jessica: Computational complexity and pragmatic solutions for flexible tile based DNA selfassembly. arXiv preprint arXiv:2108.00035 (2021) 8. Ellis-Monaghan, Jo and Pangborn, Greta and Beaudin, Laura and Miller, David and Bruno, Nick and Hashimoto, Akie: Minimal tile and bond-edge types for self-assembling DNA graphs. Discrete and Topological Models in Molecular Biology, pp. 241–270. Springer (2014) 9. P. Erd˝os, A. Rényi and V. Sós, On a problem of graph theory, Studia Sci. Math. Hungar., 1 (19661), 215–235.

The Pansophy of Semi Directed Graphs Jeffe Boats and Lazaros Kikas

Abstract Given an ordered list of randomly-selected pairs of vertices in a graph, how many of these pairs can be connected with disjoint paths? The pansophy of a graph G is the expected number of possible disjoint paths—this has been calculated and studied for many classes of undirected graphs. In this chapter we study the pansophy of various graphs where an edge or a select collection of edges have been directed. By doing this, how is the pansophy of G affected? Do specific selections of edges affect pansophy differently from other selections? These and other questions are addressed in this chapter. Keywords Interconnection networks · Graphs · Vertex disjoint paths · Pansophy · Directed graphs · Mathematical expectation

1 Introduction Let G be graph with n vertices. Suppose we choose k distinct pairs of vertices .(si , ti ) for .i = 1, . . . , k. Does there exist k vertex disjoint paths, in G, so that each path is an .si − ti path? If the answer is yes for any collection of k distinct pairs of vertices, then we say that G has k disjoint path property. In this chapter we reintroduce the concept of pansophy and how the k disjoint path property motivated the definition. We then study the pansophy of various graphs where an edge or a select collection of edges have been directed. By doing this, how is the pansophy of G affected? Do specific selections of edges affect pansophy differently from other selections? These and other questions are addressed in this chapter.

J. Boats (✉) · L. Kikas University of Detroit Mercy, Detroit, MI, USA e-mail: [email protected]; [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 S. Heuss et al. (eds.), Combinatorics, Graph Theory and Computing, Springer Proceedings in Mathematics & Statistics 462, https://doi.org/10.1007/978-3-031-62166-6_28

391

392

J. Boats and L. Kikas

2 Definition of Pansophy It was shown by Cheng et al. [1] that the alternating group graph .AGn has the .n − 2 disjoint property for .n ≥ 5. When .n = 4, it can be shown easily, by example, that .AG4 does not have the .2−disjoint path property. See [1, 3]. However, if the pairs .(s1 , t1 ) and .(s2 , t2 ) are chosen randomly, the probability that two disjoint paths can be routed is greater than 0.995. And since the alternating group graph .AG4 has 4! . 2 = 12 vertices, we can take this further. Suppose we randomly select six pairs of vertices .(s1 , t1 ), (s2 , t2 ), . . . , (s6 , t6 ). What is the probability that we can construct six disjoint paths? What is the expected number of disjoint paths that we can hope to construct when we randomly select distinct pairs of vertices? This leads to the definition of pansophy. In 2017, Boats and Kikas introduced the idea of pansophy of a graph G. [2] Let n .Ω = ⎿ ⏌ and randomly select an assignment of .Ω distinct pairs of vertices .(si , ti ) 2 for .i = 1, 2, . . . Ω. We then connect as many of these pairs as possible, in order, so that the resulting paths are vertex disjoint. The number of paths we can route is called the maximal routing volume for that assignment. We define the pansophy of the graph G, denoted .Ψ (G), to be the expected value of the maximal routing volume. For further insights into pansophy please see [4]. Let .φk (G) be the probability of being able to route, disjointly, k randomly selected distinct pairs .(si , ti ), for .i = 1, 2, 3, .., k. It was shown in [2] that we can compute .Ψ (G) by the following formula: Ψ (G) =

Ω ⎲

.

φi (G).

i=1

Example 1 Consider the cycle graph .C8 . Here .Ω = 82 = 4. Since .Cn is connected, 2 .φ1 (Cn ) = 1, and it was shown in [2] that .φk (Cn ) = (2k−1)!! , for .k = 2, . . . , Ω. We compute directly, φ1 (C8 ) = 1, φ2 (C8 ) =

.

.

2 2 2 , φ3 (C8 ) = , φ4 (C8 ) = . 3 15 105

⇒ Ψ (C8 ) =

4 ⎲

φi (C8 ) = 1.819.

i=1

3 Pansophy for Graphs with One Directed Edge Let G be a graph and let .φk (G) be the probability that a solution of k disjoint paths can be found for a randomly selected .(si , ti ) assignment. When G is connected, .φ1 (G) = 1 automatically. If we alter G by adding direction to a previously

Semi Directed

393

undirected edge e, the only way this could affect .φk is if one of the disjoint paths in the solution must necessarily use e in the opposite direction. The more direction that is added to a graph, the more deleterious the effect on pansophy. For this reason, it is clear that for any graph G and a graph .GD formed by adding direction to one or more edges, we must have .Ψ (GD ) ≤ Ψ (G). A much more interesting and difficult question is how to precisely quantify the effect. Let G be an undirected graph, and let .Ge be the graph formed by adding direction to an edge e. For any vertex assignment whose solution requires the use of e, there is a probability of . 12 that the solution will no longer be possible, due to e being oriented in the opposite direction as its path. If we define .μ(e, k) to be the probability that e must be used in the k-solution, then we obtain the result: 1 φk (Ge ) = φk (G) [1 − μ(e, k)] . 2

.

Claim For all .n ≥ 4 and .k > 1 and for any edge e in .Cn , .μ(e, k) = 12 . Proof Let .k > 1 and let a solvable assignment .A = {(s1 , t1 ), . . . , (sk , tk )} be given. The solution for A is unique, and consists of a collection of edges E which form k disjoint paths. Now consider the graph .E ∗ , the complement of E, which also consists of k disjoint paths on .Cn . The paths of .E ∗ form a unique solution to a different k-assignment of the same 2k vertices. Therefore, given any edge e, for every k-assignment solution requiring e, there is a corresponding k-solution that does not. It follows that .μ(e, k) = 12 . This enables us to compute the pansophy of .CnD1 , the cycle graph of n vertices and one directed edge. Directing only one edge leaves .CnD1 still strongly connected, so .φ1 (CnD1 ) = 1. For .n ≥ 4 and .2 ≤ k ≤ Ω, φk (CnD1 ) = φk (Cn ) [ 1 −

.

1 1 2 3 3 × ]= × = . 2 2 (2k − 1)!! 4 2(2k − 1)!! 3⎲ 1 3 φk (Cn ) = . . . = + Ψ (Cn ) . 4 4 4 Ω

.

⇒ Ψ (CnD1 ) = 1 +

k=2

Sadly, deriving explicit formulas for pansophy is seldom this simple. With .Cn , we exploited the fact that cycle graphs are Cayley graphs with both vertex-symmetry and edge-symmetry. While all Cayley graphs are vertex-symmetric, they are not all edge symmetric. The symmetry group graph .S3 provides the simplest example of how pansophy can be diminished by different amounts depending on which edge is given direction. The vertices of .S3 are permutations of 3 objects, generated by the permutations .a = (23) and .b = (123). The result is a Cayley graph where b represents movement around the 3-cycles of even or odd permutations, and a represents movement back and forth between the 3-cycles. It can be shown that the undirected graph .S3 has pansophy .Ψ (S3 ) = 11 5 .

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But the pansophy of .S3D is less: Ψ (S3D ) =

3 ⎲

.

φi (S3D ) = 1 +

i=1

7 109 7 + = = 1.817 . 10 60 60

If we consider .S3a , where only an a-edge is directed, and compare it with .S3b , we see a subtle difference in the effect on pansophy.

By brute force, we calculate: Ψ (S3a ) =

3 ⎲

.

φi (S3a ) = 1 +

i=1

7 191 8 + = = 2.122 . 9 30 90

Meanwhile the b-edge result is: Ψ (S3b ) =

3 ⎲

.

i=1

φi (S3b ) = 1 +

3 121 49 + = = 2.017 . 60 15 60

Regardless of what symmetries a graph has (or hasn’t), the direction of a single directed edge turns out to be irrelevant, as the value of .μ(e, k) is unaffected. However, the direction will become relevant when more than one edge is directed, as will the relative positions of those edges.

Semi Directed

395

4 Path Graphs with One Directed Edge Moving on to the path graphs, we find computing .μ(e, k) to be more complicated than for the cycle graphs, because .Pn is neither edge-symmetric nor vertexsymmetric. If we define .ξ(e, k) to be the number of k-solutions possible in .Pn , that is the number of .(s1 , t1 ) assignments which allow k disjoint paths, then: μ(e, k) =

.

ξ(e, k) Cn,2k

k ≤ Ω.

As for computing .ξ(e, k), this can be done combinatorially for each of the edges. Define .ej to be the j th edge from the end of the path. Then there are j vertices on one side of this edge, and .n − j vertices on the other side. Thus .ξ(ej , k) is the total number of selections of 2k vertices such that there are an odd number on either side of .ej . This can be expressed as a summation: ξ(ej , k) = Cj,1 Cn−j,2k−1 + Cj,3 Cn−j,2k−3 + . . . + Cj,2k−1 Cn−j,1 .

=

∑k

i=1 Cj,2i−1 Cn−j,2(k−i)+1

.

In the above formula, .Cα,β should be counted as zero whenever .α < β. We need look no farther than .P4 as a first example where it makes a difference which edge is directed.

On the left, .P4D1 is shown with directed edge .e1 . We can see there are three 1solutions out the six which involve this edge, and .e1 is used in the only 2-solution. Meanwhile, for .P4D2 , we see .e2 is used in four 1-solutions, and not used in the only 2-solution. This is consistent with our computations: ξ(e1 , 1) = C1,1 C3,1 = 1(3) = 3 ⇒ μ(e1 , 1) =

1 3 ξ(e1 , 1) = = C4,2 6 2

ξ(e1 , 2) = C1,1 C3,3 = 1(1) = 1 ⇒ μ(e1 , 2) =

ξ(e1 , 2) 1 = =1 C4,4 1

.

.

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J. Boats and L. Kikas

2 4 ξ(e2 , 1) = = C4,2 6 3

ξ(e2 , 1) = C2,1 C2,1 = 2(2) = 4 ⇒ μ(e2 , 1) =

.

ξ(e2 , 2) = 0

.

⇒ μ(e2 , 2) =

ξ(e2 , 2) 0 = =0 C4,4 1

Recalling from [2] that .φ1 (P4 ) = 1 and .φ2 (P4 ) = 13 , we can now compute the pansophies of .P4D1 and .P4D2 , and observe that they are different.

.

φ1 (P4D1 ) = φ1 (P4 )[1 − 12 μ(e1 , 1)] = 1[1 −

1 2

× 12 ] =

3 4

φ2 (P4D1 ) = φ2 (P4 )[1 − 12 μ(e1 , 2)] = 13 [1 −

1 2

× 1] =

1 6

⇒ Ψ (P4D1 ) =

.

∑2

D1 i=1 φi (P4 )

=

3 4

+

1 6

φ1 (P4D2 ) = φ1 (P4 )[1 − 12 μ(e2 , 1)] = 1[1 −

1 2

× 23 ] =

2 3

φ2 (P4D2 ) = φ2 (P4 )[1 − 12 μ(e2 , 2)] = 31 [1 −

1 2

× 0] =

1 3

⇒ Ψ (P4D2 ) =

∑2

D2 i=1 φi (P4 )

=

2 3

+

1 3

=

11 12

=1

5 Pansophy with Two Edges Directed Here we consider the cycle graph .C4 . We’ve shown previously that: Ψ (C4 ) = φ1 (C4 ) + φ2 (C4 ) = 1 +

.

Ψ (C4D1 ) =

.

5 2 = ; 3 3

3 1 3 1 3 5 + Ψ (C4 ) = + × = . 4 4 4 4 3 2

We now consider various versions of semi directed .C4 , where we direct exactly two edges. Before our exploration, we develop the following notation for directed edges on a cycle graph. Select a directed edge to be “Edge 1,” and label edges 2 through n by moving around the cycle in the direction of “Edge 1.” We attach a superscript to .Cn of the form “D” followed by the labels of all directed edges, placing a bar over the numbers of any edge whose direction is opposite “Edge 1.” ¯ For example, .C7D124 would be the cycle graph of seven vertices where an edge is oriented clockwise, and as we then proceed in a clockwise direction, the next edge is oriented counterclockwise, the one after that is undirected, the one after that is clockwise, and all the rest are undirected.

Semi Directed

397

To symmetry, there are five semi-directed cycle graphs of four vertices with two directed edges—we first consider the two with directed edges of the same orientation: .C4D12 and .C4D13 . Note that .φ1 (C4D12 ) = 1 since .C4D12 is strongly connected. Now we compute .φ2 . We label the vertices E, N, W and S. Depending on where .s1 is placed we can compute the probability of completing the two paths. They are: 1. 2. 3. 4.

If .s1 is at W then the probability is . 12 ; If .s1 is at N then the probability is . 31 ; If .s1 is at E, then the probability is . 16 ; If .s1 is at S, then the probability if . 31 .

Hence .φ2 (C4D12 ) = 41 ( 12 + 13 + 16 + 13 ) = 13 , the average of those four equallylikely possibilities. And so .Ψ (C4D12 ) = 1 + 31 = 43 . Now consider the graph .C4D13 . Here the edges are directed W to N and E to S. Again, since .C4D13 is strongly connected, .φ1 (C4D13 ) = 1. To compute .φ2 , we apply the same reasoning as before: 1. If .s1 is at W or E, then the probability is . 21 ; 2. If .s1 is at N or S then the probability is . 31 . 5 5 Hence .φ2 (C4D13 ) = 41 ( 12 + 13 + 12 + 13 ) = 12 , and .Ψ (C4D13 ) = 1 + 12 = 17 12 . With the directed edges sharing the same orientation, what mattered was their relative locations. But something different happens when we consider the graphs ¯ ¯ ¯ whose two directed edges are in opposing orientations: .C4D12 , .C4D13 and .C4D14 .

These graphs are not strongly-connected, so we lose the property that .φ1 is always 1. By similar reasoning we obtain:

398

J. Boats and L. Kikas ¯

3 4

and φ1 (C4D12 ) =

¯

2 3

and φ1 (C4D13 ) =

¯

3 4

and φ1 (C4D14 ) =

φ1 (C4D12 ) = .

φ1 (C4D13 ) = φ1 (C4D14 ) =

¯

1 3

⇒ Ψ (C4D12 ) =

¯

13 12

¯

5 12

⇒ Ψ (C4D13 ) =

¯

13 12

¯

1 3

⇒ Ψ (C4D14 ) =

¯

13 12

Surprisingly, the relative positions of the directed have no affect on the pansophy in this case. We do not believe, however, that this will continue to be true when our study expands to more complicated cases with more directed edges.

6 Conclusions and Open Questions This chapter was the first exploration in extending the concept of pansophy to graphs where a subset of edges were directed. We saw that by just directing a single edge, the graph’s pansophy decreased. We also saw that our choice of edges to direct also affected pansophy in different ways. However, we are also left with many open questions for future research. Among these are: 1. We considered semi directed versions of .C4 and .P4 . Derive formulas for pansophy of semi directed and fully directed versions of .Cn and .Pn as we did in [2] for .Ψ (Cn ) and .Ψ (Pn ). 2. Arrive formulas for the pansophy of directed version of .S3 and .Sn . 3. Disjoint paths were constructed from .si to .ti . Suppose that we allowed our disjoint paths to include directed paths from .ti to .si . How would this affect pansophy?

References 1. E. Cheng, L.D. Kikas, and S. Kruk. A disjoint path problem in the alternating group graph. Congressus Numerantium, 175:117–159, 2005. 2. J. Boats and L.D. Kikas. The pansophy of a graph. Congressus Numerantium, 229:125–134, 2017. 3. Lazaros D. Kikas. Interconnection networks and the k-disjoint path property. Ph.D Thesis, Oakland University, 2004. 4. Isaac Clarence Wass. A treatise on pansophy. Ph.D Thesis, Iowa State University, 2020.

Signed Magic Arrays with Certain Property Abdollah Khodkar

and David Leach

Abstract A signed magic array, .SMA(m, n; s, t), is an .m × n array with the same number of filled cells s in each row and the same number of filled cells t in each column, filled with a certain set of numbers that is symmetric about the number zero, such that every row and column has a zero sum. We use the notation .SMA(m, n) if .m = t and .n = s. In this chapter, we prove that for every even number .n ≥ 2 there exists an .SMA(m, n) such that the entries .±x appear in the same row for every .x ∈ {1, 2, 3, . . . , mn/2} if and only if (a) .n = 2 and .m ≡ 0, 3 (mod 4), or (b) .n ≥ 4 and .m ≥ 3. Keywords Magic array · Heffter array · Signed magic array · Shiftable array

1 Introduction An integer Heffter array .H (m, n; s, t) is an .m × n array with entries from .X = {±1, ±2, .. . . , ±ms} such that each row contains s filled cells and each column contains t filled cells, the elements in every row and column sum to 0 in .Z, and for every .x ∈ X, either x or .−x appears in the array. The notion of an integer Heffter array .H (m, n; s, t) was first defined by Archdeacon in [1]. A Heffter array is tight if it has no empty cell; that is, .n = s (and necessarily .m = t). Integer Heffter arrays with .m = n = s represent a type of magic square where each number from the set 2 .{1, 2, 3, . . . , n } is used once up to sign. The proof of the following Theorem can be found in [2].

A. Khodkar · D. Leach (✉) Department of Computing and Mathematics, University of West Georgia, Carrollton, GA, USA e-mail: [email protected]; [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 S. Heuss et al. (eds.), Combinatorics, Graph Theory and Computing, Springer Proceedings in Mathematics & Statistics 462, https://doi.org/10.1007/978-3-031-62166-6_29

399

400 Fig. 1 An .SMA(3, 2), and .SMA(4, 2)

.SMA(3, 4)

A. Khodkar and D. Leach 1 1 −1 1 −1 2 −2 −2 2 −2 5 4 −5 −4 −3 −3 3 −6 −3 3 6 4

−1 2 3 −4

Theorem 1 Let .m, n be integers at least 3. There is a tight integer Heffter array if and only if .mn ≡ 0, 3 (mod 4). For more information on Integer Heffter arrays consult [1–4]. A signed magic array .SMA(m, n; s, t) is an .m × n array with entries from X, where .X = {0, ±1, ±2, ±3, . . . , ±(ms − 1)/2} if ms is odd and .X = {±1, ±2, ±3, . . . , ±ms/2} if ms is even, such that precisely s cells in every row and t cells in every column are filled, every integer from set X appears exactly once in the array and the sum of each row and of each column is zero. An .SMA(m, n; s, t) is called tight, and denoted .SMA(m, n), if it contains no empty cells; that is .m = t (and necessarily .n = s). Figure 1 displays tight .SMA(3, 2), .SMA(3, 4) and .SMA(4, 2). An .SMA(m, n; s, t) is shiftable if it contains the same number of positive as negative entries in every column and in every row. These arrays are called shiftable because they may be shifted to use different absolute values. By increasing the absolute value of each entry by k, we add k to each positive entry and .−k to each negative entry. If the number of entries in a row is .2𝓁, this means that we add .𝓁k + 𝓁(−k) = 0 to each row, and the same argument applies to the columns. Thus, when shifted, the array retains the same row and column sums. The proof of the following theorem can be found in [8]. Theorem 2 An .SMA(m, n) exists precisely when .m = n = 1, or when .m = 2 and n ≡ 0, 3 (mod 4), or when .n = 2 and .m ≡ 0, 3 (mod 4), or when .m, n > 2.

.

Corollary 1 There exists an .SMA(m, 2) such that the entries .±x are in the same row for every .x ∈ {1, 2, 3, . . . , m} if and only if .m ≡ 0, 3 (mod 4). We also note that if A is an .m × n tight integer Heffter array, then the .m × 2n array .[A, −A] is an .SMA(m, 2n) with the property that the entries .±x appear in the same row for every F.x ∈ {1, 2, . . . , mn}. For more information on signed magic arrays consult [5–8]. In this chapter, we prove that for every even number .n ≥ 2 there exists an .SMA(m, n) such that the entries .±x appear in the same row for every .x ∈ {1, 2, 3, . . . , mn/2} if and only if (a) .n = 2 and .m ≡ 0, 3 (mod 4), or (b) .n ≥ 4 and .m ≥ 3. For simplicity, we say an .SMA(m, n), with n even, has the required property if the entries .±x appear in the same row for every .x ∈ {1, 2, 3, . . . , mn/2}.

Signed Magic Arrays with Certain Property

401

2 The Case m and n Are Even Theorem 3 Let m, n ≥ 4 and even. Then there exists a shiftable SMA(m, n) with the required property. Proof Proceed by strong induction first on n and then on m. As the base case, we provide arrays for (m, n) = (4, 4), (6, 4), (4, 6) and (6, 6) in Figs. 2 and 3, respectively. Now, let m ∈ {4, 6} and n be even, and assume that there exists a shiftable SMA(m, n − 4) with the required property. We may extend this array by adding four columns to create an m × n array. The empty m × 4 array may be filled by a shifted copy of the SMA(4, 4) or SMA(6, 4) in Fig. 2. As the shifted copies each has a row and column sum of zero, they do not change the row sums from the m × (n − 4) array, and the sums of the new columns will be zero as well. Moreover, the shifted copies have the required property. Therefore, a shiftable SMA(m, n) exists with the required property. Hence, by strong induction on n, a shiftable SMA(m, n) with the required property exists for m ∈ {4, 6} and n ≥ 4 even. See Fig. 4 for an illustration. Now, let m and n both be even, and m, n ≥ 4. Assume that there exists a shiftable SMA(m − 4, n) with the required property. We may extend this array by adding four rows to create an m × n array. The empty 4 × n array may be filled by a shifted copy of a shiftable SMA(4, n) with the required property, which exists by the above argument. Hence, by strong induction on n, a shiftable SMA(m, n) with the required property exists for m, n ≥ 4 and even.

Fig. 2 SSMA(4, 4) and SSMA(6, 4) with the required property

Fig. 3 SSMA(4, 6) and SSMA(6, 6) with the required property

1 −2 −3 4

−1 2 3 −4

1 −2 −3 4

−5 6 7 −8

5 −6 −7 8

−1 2 3 −4

−9 9 10 − 10 11 − 11 − 12 12

5 −6 −7 8

−5 6 7 −8

1 2 3 −5 6 −7

−1 −2 −3 5 −6 7

2 4 −1 −3 5 −7 4 −8 −9 11 − 13 15

−2 −4 1 3 −5 7

6 −8 12 −9 10 − 11

−6 8 − 12 9 − 10 11

−4 8 10 − 16 17 − 15

12 14 − 10 − 11 13 − 18

− 12 − 14 9 16 − 17 18

402

A. Khodkar and D. Leach 1 2 3 −5 6 −7

Fig. 4 SSMA(6, 10) with the required property obtained by constructions given in Theorem 3

−1 −2 −3 5 −6 7

4 −8 −9 11 − 13 15

−4 8 10 − 16 17 − 15

12 14 − 10 − 11 13 − 18

− 12 20 − 20 − 14 22 − 22 9 − 19 19 16 − 21 21 − 17 23 − 23 18 − 25 25

24 − 26 30 − 27 28 − 29

− 24 26 − 30 27 − 28 29

3 The Case m Odd and n Even Since the structure of the .SMA(3, n) given below is crucial in our constructions, we include the proof of this lemma here which can also be found in [8]. Lemma 1 Let n be even. Then there exists an .SMA(3, n) with the required property. Proof An .SMA(3, 2) and an .SMA(3, 4) are given in Fig. 1. Now let .n = 2k ≥ 6 and .pj = ⎾ j2 ⏋ for .1 ≤ j ≤ 2k . Define a .3 × n array .A = [ai,j ] as follows: For .1 ≤ j ≤ 2k,

a1,j

.

⎞ ⎧ ⎛ 3pj −2 ⎪ − j ≡0 ⎪ 2 ⎪ ⎪ ⎪ ⎨ 3pj −1 ⎞ j ≡1 ⎛2 = 3pj −1 ⎪ j ≡2 − ⎪ ⎪ 2 ⎪ ⎪ 3p −2 ⎩ j j ≡3 2

(mod 4) (mod 4) (mod 4) (mod 4).

For the third row we define .a3,1 = −3k, .a3,2k = 3k and when .2 ≤ j ≤ 2k − 1

a3,j

.

⎧ ⎪ −3(k − pj ) ⎪ ⎪ ⎪ ⎨3(k − p + 1) j = ⎪ −3(k − pj ) ⎪ ⎪ ⎪ ⎩ 3(k − pj + 1)

j ≡ 0 (mod 4) j ≡ 1 (mod 4) j ≡ 2 (mod 4) j ≡ 3 (mod 4).

Finally, .a2,j = −(a1,j + a3,j ) for .1 ≤ j ≤ 2k. It is straightforward to see that array A is an .SMA(3, n) with the required property. Figure 7 in Appendix 1 displays an .SMA(3, 12) constructed by the above method. Lemma 2 Let .n ≥ 4 and even. Then there exists an .SMA(5, n) with the required property. Proof We consider two cases. Case 1: .n ≡ 0 (mod 4) Let A be an .SMA(3, n) constructed by Lemma 1. By construction, the entries in row one of A are

Signed Magic Arrays with Certain Property Fig. 5 SMA(5, 4); constructed by a .(5, 3) Heffter array and its opposite .SMA(5, 6)

403 −9 9 7 10 − 10 − 8 1 −1 2 4 5 −4 −6 −3 3

− 7 − 14 12 2 14 − 12 8 −6 −1 7 6 1 − 2 4 − 13 9 − 4 13 − 5 11 − 8 − 3 − 11 8 6 5 10 − 15 − 5 − 10

−2 −7 −9 3 15

{±(3i + 1), ±(3i + 2) | 0 ≤ i ≤ (n − 4)/4} (See Fig. 7 in Appendix 1).

.

We rearrange the entries of row one as follows: For .0 ≤ i ≤ (n − 4)/4, switch the positions of entry .3i + 1 with entry .3i + 2, and also switch the positions of entry .−(3i +1) with entry .−(3i +2). Call this rearranged array B. See Fig. 8 in Appendix 1. Note that the row sum is still zero in B and the column sums consists of .n/2 ones and .n/2 negative ones. We now extend array B by adding two rows to create a .5 × n array C. Let j .qj = ⎿ ⏌, and define the entries of rows four and five as 2 c4,j = (−1)qj (3k + 2pj − 1), and c5,j = (−1)qj +1 (3k + 2pj ).

.

In the resulting array C all the columns and rows sum to zero. See Fig. 9 in Appendix 1. Case 2: .n ≡ 2 (mod 4) Figure 5 displays an .SMA(5, 6) with the required property. Let .n ≥ 10 and let A be an .SMA(3, n) constructed by Lemma 1. By construction, the numbers in row one of A are: {±(3i + 1), ±(3i + 2) | 0 ≤ i ≤ (n − 6)/4} ∪ {±(3n − 2)/4}.

.

See Fig. 10 in Appendix 2. We now rearrange the entries of row one as follows: For .0 ≤ i ≤ (n − 10)/4, switch the positions of entry .3i + 1 with entry .3i + 2, and switch the position of entry .−(3i + 1) with entry .−(3i + 2). Also switch the positions of entry . 3(n−6) +1 4 3(n−6) with entry . 4 + 2. Note that we switch every entry in row one except the entries in columns .n − 4, n − 2, n − 1 and n. Now we switch the entries in columns .n − 4, n − 2, n − 1 and n of row two as follows: switch . −(3n+2) with . −(3n+10) and switch . 3n+2 with . 3n+10 4 4 4 4 . Call the resulting .3 × 10 array B. See Fig. 11 in Appendix 2. Note that the row sum is still n−4 zero in B and the column sums consists of . n−4 2 ones, . 2 negative ones, 2 twos and 2 negative twos. We now extend array B by adding two rows to create a .5 × n array C. Let j .qj = ⎿ ⏌, and define the entries of row four by 2

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c4,j =

.

⎧ ⎪ (−1)qj +1 (3k + 2pj − 1) ⎪ ⎪ ⎪ ⎪ ⎪ −(5k − 3) ⎪ ⎪ ⎪ ⎨−(5k − 5) ⎪ 5k − 3 ⎪ ⎪ ⎪ ⎪ ⎪ 5k − 2 ⎪ ⎪ ⎪ ⎩ −(5k − 2)

1≤j ≤n−5 j =n−4 j =n−3 j =n−2 j =n−1 j = n.

The entries of row five are determined, and in the resulting array C, all the columns and rows sum to zero. See Fig. 12 in Appendix 2. Theorem 4 Let .m ≥ 3 be odd and n be even. There exists an .SMA(m, n) with the required property if and only if (a) .n = 2 and .m ≡ 0, 3 (mod 4), or (b) .n ≥ 4 and .m ≥ 3. Proof For .n = 2 we apply Corollary 1. Now let .m ≥ 3 and .n ≥ 4. We consider two cases. Case 1: .m ≡ 3 (mod 4) By Lemma 1 the statement is true for .m = 3. Let .m ≥ 7 and let A be an .SMA(3, n) constructed in Lemma 1. We extend array A by adding .m − 3 rows to create an .m × n array. The empty .(m − 3) × n array may be filled by a shifted copy of the .SMA(m − 3, n) given by Theorem 3. As the shifted copies each have a row and column sum of zero, they do not change the row sums from the .3 × n array, and the sums of the new columns will be zero as well. Moreover, the shifted copies have the required property. Therefore, an .SMA(m, n) exists with the property that the entries .±x appear in the same row for every .x ∈ {1, 2, 3, . . . , mn/2}. Case 2: .m ≡ 1 (mod 4) By Lemma 2 the statement is true for .m = 5. Now let .m ≥ 9 and let A be an .SMA(5, n) constructed in Lemma 2. We now extend array A by using essentially the same construction as in case 1, but adding .m − 5 rows to A and filling the empty rows using shifted copies of the .SMA(m − 5, n) from Theorem 4. This gives an .SMA(5, n) with the required property (Fig. 6). Main Theorem For every even number .n ≥ 2 there exists an .SMA(m, n) such that the entries .±x appear in the same row for every .x ∈ {1, 2, 3, . . . , mn/2} if and only if (a) .n = 2 and .m ≡ 0, 3 (mod 4), or (b) .n ≥ 4 and .m ≥ 3. Fig. 6 .SMA(5, 8) obtained from .H (5, 4)

2 9 7 1 −2 − 8 12 13 4 8 − 17 3 − 16 − 6 17 18 − 14 11 − 19 − 18 5 − 10 − 15 20 − 5

−9 − 12 −3 14 10

−7 − 13 16 − 11 15

−1 −4 6 19 − 20

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Appendix 1: An Example for Lemma 2 Case 1 See Figs. 7, 8, and 9. 1 −1 2 −2 4 −4 5 −5 7 −7 8 −8 17 16 − 17 14 − 16 13 − 14 11 − 13 10 − 11 − 10 − 18 − 15 15 − 12 12 − 9 9 − 6 6 − 3 3 18

Fig. 7 The .SMA(3, 12) denoted by A, constructed by Lemma 1

2 17 − 18 1

−2 1 −1 5 16 − 17 14 − 16 − 15 15 − 12 12 −1 −1 1 1

−5 4 13 − 14 −9 9 −1 −1

−4 8 11 − 13 −6 6 1 1

−8 7 −7 10 − 11 − 10 − 3 3 18 −1 −1 1

Fig. 8 The .3 × 12 array B, obtained from the .SMA(3, 12) given above, using the construction given in Lemma 2; Row 4 displays the column sums

2 17 − 18 19 − 20

−2 16 − 15 − 19 20

1 − 17 15 − 21 22

−1 14 − 12 21 − 22

5 − 16 12 23 − 24

−5 4 13 − 14 −9 9 − 23 − 25 24 26

−4 8 11 − 13 −6 6 25 27 − 26 − 28

−8 7 10 − 11 −3 3 − 27 − 29 28 30

−7 − 10 18 29 − 30

Fig. 9 The SMA(5, 12) denoted by C, obtained by the construction given in Lemma 2, Case 1

Appendix 2: An Example for Lemma 2 Case 2 See Figs. 10, 11, and 12. 1 −1 2 −2 4 −4 5 −5 7 −7 14 13 − 14 11 − 13 10 − 11 8 − 10 − 8 − 15 − 12 12 − 9 9 − 6 6 − 3 3 15

Fig. 10 The .SMA(3, 10), denoted by A, constructed by Lemma 1

2 14 − 15 1

−2 1 13 − 14 − 12 12 −1 −1

−1 5 11 − 13 −9 9 1 1

−4 4 8 − 11 −6 6 −2 −1

−5 10 −3 2

7 −8 3 2

−7 − 10 15 −2

Fig. 11 The .3 × 10 array B obtained from the .SMA(3, 10) given above, using the construction given in Lemma 2; Row 4 displays the column sums

406 Fig. 12 The .SMA(5, 10), denoted by C, obtained by the construction given in Lemma 2, Case 2

A. Khodkar and D. Leach 2 14 − 15 16 − 17

−2 13 − 12 − 16 17

1 − 14 12 − 18 19

−1 5 11 − 13 −9 9 18 20 − 19 − 21

−4 4 8 − 11 −6 6 − 22 − 20 24 21

−5 7 10 − 8 −3 3 22 23 − 24 − 25

−7 − 10 15 − 23 25

References 1. D. S. Archdeacon, Heffter arrays and biembedding graphs on surfaces, Electron. J. Combin. 22 (2015), #P1.74. 2. D. S. Archdeacon, T. Boothby and J. H. Dinitz, Tight Heffter arrays exist for all possible values, J. Combin. Des. 25 (2017), 535. 3. D. S. Archdeacon, J. H. Dinitz, D. M. Donovan, and E. S. Yazici, Square integer Heffter arrays with empty cells, Des. Codes Cryptogr. 77 (2015), 409–426. 4. J. H. Dinitz and I. M. Wanless, The existence of square integer Heffter arrays, Ars Math. Contemp. 13 (2017), 81–93. 5. A. Khodkar, D. Leach and B. Ellis, Signed magic rectangles with three filled cells in each column, Bulletin of the Institute of Combinatorics and its Applications, 90 (2020), 87–106. 6. A. Khodkar and D. Leach, Magic Squares with empty cells, Ars Combinatoria 154 (2021), 45–52. 7. A. Khodkar and B. Ellis, Signed magic rectangles with two filled cells in each column, J. Combin. Math. Combin. Comput. 121 (2024), 31–40. 8. A. Khodkar, C. Schulz and N. Wagner, Existence of Some Signed Magic Arrays, Discrete Math. 340 (2017), 906–926.

The Buratti-Horak-Rosa Conjecture Holds for Some Underlying Sets of Size Three Pranit Chand and M. A. Ollis

Abstract The Buratti-Horak-Rosa Conjecture concerns the possible multisets of edge-labels of a Hamiltonian path in the complete graph with vertex labels .0, 1, . . . , v − 1 under a particular induced edge-labeling. The conjecture has been shown to hold when the underlying set of the multiset has size at most 2, is a subset of .{1, 2, 3, 4} or .{1, 2, 3, 5}, or is .{1, 2, 6}, .{1, 2, 8} or .{1, 4, 5}, as well as partial results for many other underlying sets. We use the method of growable realizations to show that the conjecture holds for each underlying set .U = {x, y, z} when .max(U ) ≤ 7 or when .xyz ≤ 24, with the possible exception of .U = {1, 2, 11}. We also show that for any even x the validity of the conjecture for the underlying set .{1, 2, x} follows from the validity of the conjecture for finitely many multisets with this underlying set.

1 Introduction The setting for the Buratti-Horak-Rosa (BHR) Conjecture is the complete graph .Kv on vertex set .{0, 1, . . . , v − 1}. In this graph, define an induced edge-labeling by 𝓁(x, y) = min(|x − y|, v − |x − y|),

.

called the length of the edge between x and y. If the vertices are arranged in a circle with the natural order, then the length of the edge between x and y corresponds to the smallest number of steps around the circle from x to y.

The authors “Pranit Chand and M. A. Ollis” contributed equally to this work. P. Chand · M. A. Ollis (✉) Marlboro Institute for Liberal Arts and Interdisciplinary Studies, Emerson College, Boston, MA, USA e-mail: [email protected]; [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2024 S. Heuss et al. (eds.), Combinatorics, Graph Theory and Computing, Springer Proceedings in Mathematics & Statistics 462, https://doi.org/10.1007/978-3-031-62166-6_30

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The edges of a subgraph .𝚪 of .Kv give a multiset of lengths in the range .1 ≤ 𝓁(x, y) ≤ ⎿v/2⏌. If a multiset L of edges arises in this way from a Hamiltonian path .𝚪, then .𝚪 realizes L or .𝚪 is a realization of L . We are interested in determining which multisets L are realizable. Example 1 Let .L = {2, 48 , 53 } (where exponents indicate multiplicity in a multiset). The Hamiltonian path [6, 10, 5, 1, 9, 11, 2, 7, 3, 12, 8, 4, 0]

.

in .K13 has edge-lengths 4, 5, 4, 5, 2, 4, 5, 4, 4, 4, 4, 4

.

and so realizes L. The BHR Conjecture—originally posed by Buratti for prime v [12], generalized to arbitrary v by Horak and Rosa [4], and given an equivalent more succinct formulation by Pasotti and Pellegrini [9]—is as follows: Conjecture 2 (BHR Conjecture) For any multiset L of size .v − 1 with underlying set .U ⊆ {1, . . . , ⎿v/2⏌}, there is a realization of L in .Kv if and only if for any divisor d of v the number of multiples of d in L is at most .v − d. The condition on divisors in the conjecture is necessary [4]. Call a multiset L whose largest element is at most .⎿v/2⏌ and that meets this necessary condition admissible. Denote the BHR Conjecture for L by .BHR(L). The BHR Conjecture has been completely solved for very few underlying sets. Two early papers independently covered those of size at most 2 [3, 4], followed shortly afterwards by a proof for .{1, 2, 3}, the first three element set to be solved [2]. Subsequent work covers subsets of .{1, 2, 3, 4} [7] or .{1, 2, 3, 5} [9], .{1, 2, 6} and .{1, 2, 8} [8], and .{1, 4, 5} [7]. In addition to this, there are strong partial results for a wide variety of underlying sets. See [7, Theorem 1.2] for a summary of the position as of mid-2021. We shall use one of these partial results in Sects. 4 and 5: when x is even, the multiset .{1a , 2b , x c } is realizable if .a + b ≥ x − 1 [8]. The BHR Conjecture fits into a network of related problems, often related to graph decompositions of various types, for example the Seating Couples Around the King’s Table problem [10] and determining whether particular Cayley graphs of the cyclic group admit cyclic decompositions [8]. Seamone and Stevens [11] ask a more general question that encompasses arbitrary groups and spanning trees, for which the specialization to the cyclic group and Hamiltonian paths is equivalent to the BHR Conjecture. Perhaps the oldest instance of a problem of this type can be connected to the Walecki Construction of 1892, see [1], which may be viewed as a verification of the BHR Conjecture when the multiset L is .{12 , 22 , . . . , (m−1)2 , m} or .{12 , 22 , . . . , m2 }. See [6] for further discussion and references concerning these connections.

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In the next section we introduce the notion of a growable realization, as developed in [7], which is our primary tool, and prove some new technical results providing constraints on when they can∏exist. For a given underlying set U , the approach divides the work into .π(U ) = x∈U x cases. In Sect. 3 we describe methods that each take a finite list of growable realizations for a multiset with underlying set U to completely prove the BHR Conjecture for one of the .π(U ) cases. We use these methods to complete the proof of the BHR Conjecture when the underlying set has size 3 and the largest element is at most 5. In Sect. 4 we show that realizations with some parameters must be growable. This lets us build on the work of Pasotti and Pellegrini in [8] for the problem with .U = {1, 2, x} for some even x. For a fixed x in this instance, we show that if the BHR Conjecture fails, then it must fail for a multiset .L = {1a , 2b , x c } of size .v − 1 with 2 .a + b < x − 1 and .v < x − 1. This is used to prove the conjecture for underlying sets .{1, 2, 10} and .{1, 2, 12}. In Sect. 5 we develop and implement an algorithm for generating a finite set of realizations to prove one of the .π(U ) cases. Using this we are able to prove the conjecture for a variety of three-element underlying sets. Taking the results of these approaches together, we prove Theorem 3 in Sect. 6. Theorem 3 Let U be a set of size 3 with .max (U ) ≤ 7 or .π(U ) ≤ 24. With the possible exception of .U = {1, 2, 11}, the BHR Conjecture holds for multisets with underlying set U . Existing results (stated previously) give 10 instances of underlying sets U of size 3 for which the BHR Conjecture is known to hold. Theorem 3 adds a further 27 underlying sets to this list.

2 Growable Realizations If a multiset L has an “x-growable” realization for L, then we can use it to produce a realization for .L ∪ {x kx } for any .k ≥ 1. When a realization is “x-growable” for multiple values of x, this is a powerful tool for constructing realizations for a wide variety of multisets. In this section we give the necessary definitions and constructions that we use. Proofs of their correctness may be found in [7]. The method works via an embedding of .Kv into .Kv+x , for some .x ≤ v/2. For some fixed .m ∈ {0, 1, . . . , v − 1}, define the growth embedding by ⎧ y |→

.

y when y ≤ m, y + x otherwise.

Let .h = [h1 , . . . , hv ] be a realization of a multiset L. Take x and m as in the growth embedding. If each y with .m − x < y ≤ m is incident with exactly one edge whose length is increased by the growth embedding and there are no other

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edges whose lengths are increased, then say that .h is x-growable at m. Let .X = {x1 , x2 , . . . , xk }. If .h is .xi -growable for each .xi ∈ X then say that .h is X-growable. Theorem 4 ([7]) Suppose a multiset L has an X-growable realization. Then for each .x ∈ X, the multiset .L ∪ {x x } has an X-growable realization. Proof Construction Let .g = [g1 , . . . , gv ] be an X-growable realization of a multiset L. Take .x ∈ X with .g x-growable at m. Apply the growth embedding to obtain a (non-Hamiltonian) path .h' in .Kv+x . Each vertex y with .m − x < y ≤ m is adjacent to exactly one vertex z in the original realization that is mapped to .z + x, meaning that y is adjacent to .z + x in .h' . Insert the vertex .y +x between them for each of these y values to give a Hamiltonian path .h in .Kv+x . The Hamiltonian path .h is the required X-growable realization of x .L ∪ {x }. ⨆ ⨅ Repeated applications of Theorem 4 gives the following corollary. Corollary 5 ([7]) Let .X = {x1 , x2 , . . . , xk }. If a multiset L has an X-growable realization, then the multiset .L ∪ {x1𝓁1 x1 , x2𝓁2 x2 , . . . , x2𝓁2 x2 } has an X-growable realization for any .𝓁1 , 𝓁2 , . . . , 𝓁k ≥ 0. Example 6 The realization for .{2, 48 , 53 } in Example 1 is .{4, 5}-growable. It is 4-growable at 3, 4 and 5 and 5-growable at 6. If we apply Theorem 4 twice, first for 4-growability (at 3) and second for 5-growability (at 10, the point to which the growth embedding moves the 5-growability value), we obtain the .{4, 5}-growable realization [10, 15, 19, 14, 9, 5, 1, 18, 20, 2, 6, 11, 16, 12, 7, 3, 21, 17, 13, 8, 4, 0]

.

for .{2, 412 , 58 }. It is 4-growable at 3 (and other values) and 5-growable at 10 (and other values). The following lemma gives some instances of admissible multisets that do not have particular growable realizations. Lemma 7 Let .L = {x a , y b , zc } with .x < y < z. Set .v = a + b + c + 1 and .X ⊆ {x, y, z}. There is no X-growable realization for L in the following situations: 1. 2. 3. 4.

z ∈ X and .v = 2z, z ∈ X and .c < z − y, .x ∈ X and .a + b < z − x − 1, .y ∈ X and .a + b < z − y − 1. . .

Proof Consider Item 1. Suppose .v = 2z and .h is a realization that is z-growable at m. To satisfy the definition, the vertex .m − z + 1 must be connected to a vertex with label at least .m + 1 and this must give an edge that is lengthened by the growth embedding. However, as .v = 2z, such edges are not lengthened by the growth embedding so we cannot have a z-growable realization.

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Now consider Item 2. Suppose .c < z − y and we have a realization .h that is zgrowable at m. The vertices .m − z + 1 to .m − y must each be incident with an edge that is lengthened, the only option for which is one of length z. However there are .z − y > c vertices in the list, so this is impossible. For Item 3, suppose .a + b < z − x − 1 and .h is a realization that is x-growable at m. Each edge of length z joins two vertices p and .p + z and x-growability means that we cannot have .m − z < p < m − x + 1. Hence .c ≤ v − z + x. Substituting .v = a + b + c + 1 gives the contradictory .a + b ≥ z − x − 1. The argument for Item 4 is the same, with y in place of x. ⨆ ⨅ Lemma 7 is useful in future sections for avoiding fruitless searches for particular growable realizations.

3 Underlying Sets with Largest Element 5 In this section we describe in more detail some possible ways we might move from a small list of growable realizations to a complete solution for one of the .π(U ) cases. It is often the case that a method misses some small subcases. The following result, proved by a computation of Meszka, deals with this situation. Theorem 8 ([5]) Let L be an admissible multiset of size .v − 1 with underlying set U . If .v ≤ 19 or .v = 23 then .BHR(L) holds. Throughout this section, fix .U = {x, y, z} and .L = {x a , y b , zc }. Corollary 5 divides the problem naturally into .π(U ) = xyz cases. To facilitate discussion of this, as in [7] we write (r1 , r2 , r3 ) ≡ (s1 , s2 , s3 ) (mod (t1 , t2 , t3 ))

.

to mean that .ri ≡ si (mod ti ) for .1 ≤ i ≤ 3. We further write (r1 , r2 , r3 ) ⪯ (s1 , s2 , s3 )

.

to mean that .ri ≤ si for .1 ≤ i ≤ 3. Let .a0 be in the range .0 < a0 ≤ x with .a0 ≡ a (mod x). Define .ai = a0 + ix for all .i > 0. Make similar definitions for .b0 and .bi with respect to y and for .c0 and .ci with respect to z. Depending which small growable realizations exist within a given case, there are various methods that we might deploy to prove the BHR Conjecture for that case. For each method the goal is the same: Produce a finite set H of realizations for which ' ' ' given any .L = {x a , y b , zc } in that case there is an .h ∈ H that realizes .{x a , y b , zc } ' ' ' ' ' with the properties that .(a , b , c ) ⪯ (a, b, c) and if .a /= a (respectively .b /= b or ' .c /= c ) then .h is x-growable (respectively y- or z-growable).

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We describe the methods in approximate increasing order of number of realizations needed (it can only be approximate as some methods have a variable number of required realizations). Some earlier methods are special cases of later ones, but we separate them out as the special cases we list are both simpler and more frequently used. Note that choosing different orderings of x, y and z gives alternative ways to implement many of the methods. The choice described corresponds to the most frequently used configuration when .x < y < z. For brevity and ease of reading, we denote the multiset .L = {x ai , y bj , zck } by the triple .(ai , bj , ck ). Method 1 Find a U -growable realization .h1 for .(a0 , b0 , c0 ). Then .h1 covers all (a, b, c) in this case with .(a, b, c) ⪰ (a0 , b0 , c0 ), which is all of them.

.

Method 2 Find a U -growable realization .h1 for .(a0 , b0 , c1 ) and a .{x, y}-growable realization .h2 for .(a0 , b0 , c0 ). The realization .h1 covers all subcases with .c /= c0 and .h2 covers those with .c = c0 . Method 3 Find a U -growable realization .h1 for .(a0 , b0 , c1 ) and .{x, y}-growable realizations .h2 and .h3 for .(ai , b0 , c0 ) and .(a0 , bj , c0 ) respectively, where .i, j > 0. The realization .h1 covers all subcases with .c /= c0 , .h2 covers all subcases with .c = 0 and .a ≥ ai , and .h3 covers all subcases with .c = 0 and .b ≥ bj . Hence we have only finitely many exceptions: those with .(a, b, c) ⪯ (ai − x, bj − y, c0 ). The largest of these has .v = ai + bj + c0 − x − y + 1. In all instances we consider, these exceptions are covered by Theorem 8. If .(a0 , b0 , ck ) is inadmissible for all .ck , then we may replace .h1 with .h4 and .h5 for .(a1 , b0 , c1 ) and .(a0 , b1 , c1 ) respectively. This covers all subcases with .c ≥ ck . Method 4 Find a U -growable realization .h1 for .(a0 , b0 , ck ) and for each .𝓁 with 0 ≤ 𝓁 < k find .{x, y}-growable realizations .h2,𝓁 and .h3,𝓁 for .(ai𝓁 , b0 , c𝓁 ) and .(a0 , bj𝓁 , c𝓁 ) respectively, where .i𝓁 , j𝓁 > 0. The realization .h1 covers all subcases with .c ≥ ck . Then .h2,𝓁 covers all subcases with .c = c𝓁 and .a ≥ ai𝓁 and .h3,𝓁 covers all subcases with .c = c𝓁 and .b ≥ bj𝓁 . Let .i = max𝓁 (i𝓁 ) and .j = max𝓁 (j𝓁 ). We have only finitely many exceptions as any exception must have .(a, b, c) ⪯ (ai − x, bj − y, ck − z). The largest of these has .v = ai + bj + ck − x − y − z + 1. In all instances we consider, these exceptions are covered by Theorem 8. .

Method 5 Find a U -growable realization .h1 for .(a1 , b1 , c1 ). Find realizations .h2 , h3 and .h4 for .(ai , b0 , c0 ) .(a0 , bj , c0 ) and .(a0 , b0 , ck ) that are x-, y- and z-growable respectively, unless the subcase is always inadmissible. Find a .{y, z}-growable realization .h5 for .(a0 , b1 , c1 ), find an .{x, z}-growable realization .h6 for .(a1 , b0 , c1 ), and find an .{x, y}-growable realization .h7 for .(a1 , b1 , c0 ). If .a = a0 and .b = b0 then either .(a, b, c) ⪰ (a0 , b0 , ck ), and so is covered by .h4 , or has .v ≤ a0 + b0 + ck − z + 1. In all cases we consider, these potential small exceptions are covered by Theorem 8. A similar argument covers the other subcases when two of .a, b, c are as small as possible. Now suppose that .a = a0 but .b > b0 and .c > c0 . Then .(a, b, c) ⪰ (a0 , b1 , c1 ) and so .h5 suffices. A similar argument

.

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covers the other subcases when exactly one of .a, b, c is as small as possible. Finally, if none of .a, b, c are as small as possible, then .(a, b, c) ⪰ (a1 , b1 , c1 ) and the subcase is covered by .h1 . While more general methods may certainly be devised and will be necessary for some underlying sets, these methods are sufficient to prove the main result of this section. Theorem 9 If .U = {2, 4, 5} or .{3, 4, 5} then .BHR(L) holds for multisets L with underlying set U . Proof For each U we give tables of realizations which, in combination with the methods described in this section and Theorem 8 imply the result. Tables 1, 2, 3, and 4 cover .U = {2, 4, 5} and Tables 5, 6, 7, and 8 cover .U = {3, 4, 5}. Each problem has .π(U ) cases, which is 40 and 60 for .{2, 4, 5} and .{3, 4, 5} respectively, and the tables cover these cases in lexographic order of .(a0 , b0 , c0 ). Note that this means that the cases that are more difficult (and hence require more intricate methods) tend to be earlier in the tables. ⨆ ⨅ Corollary 10 Suppose .|U | = 3 and .max (U ) ≤ 5. Then .BHR(L) holds for multisets with underlying set U . Proof It is known that .BHR(L) holds when the underlying set of L is a subset of {1, 2, 3, 4} or .{1, 2, 3, 5} [7, 9]. Hence we may assume .U = {x, 4, 5} with .x ≤ 3. The case .{1, 4, 5} is covered in [7] and .{2, 4, 5} and .{3, 4, 5} are covered by Theorem 9. ⨆ ⨅

.

4 Underlying Set {1, 2, x} In this section we study underlying sets of the form .{1, 2, x} with x even. However, before narrowing our focus, we give the more general result that is behind the work of this section. Lemma 11 Let U be a set with .max (U ) = μ. Let L be an admissible multiset of size .v − 1 with underlying set U and suppose x appears at least .(μ − 1)v/μ times in L. Then a realization of L is necessarily x-growable. Proof Suppose that there are .μ consecutive vertices .hi+1 , hi+2 , . . . , hi+μ in a realization of L that are each incident with two edges of length x. Then the growth embedding at .m = i + μ using x lengthens exactly one of the edges incident with each of the vertices .hi+μ−x+1 , . . . , hi+μ . Any other edge that is lengthened must have length greater than .μ, and so there are none and the realization is x-growable at .i + μ. Within the Hamiltonian path that forms the realization, each vertex is incident with one or two edges. Considering the vertices adjacent to just one edge as being adjacent to that edge and an edge of length 0, there are 2v incidences in total. At

414

P. Chand and M. A. Ollis

Table 1 Growable realizations for .{2a , 4b , 5c } separated by congruence modulo .(2, 4, 5): case to .(1, 2, 3). They are x-growable at .mx in .m = (m2 , m4 , m5 )

.(1, 1, 1)

Case (a,b,c) (1,1,1) (3, 5, 6) (7, 1, 1) ( 1, 1, 11) (1, 5, 6 ) (3, 1, 6) (3, 5, 1) (1,1,2) ( 3, 5, 7 ) ( 7, 1, 2 ) ( 1, 9, 2 ) ( 1, 5, 7 ) ( 3,1, 7 ) ( 3, 5, 2 ) (1,1,3) (3, 5, 8 ) ( 5,1, 3 ) ( 1, 5, 3 ) ( 1, 1, 8 ) ( 1, 5, 8 ) ( 3, 1, 8 ) ( 3, 5, 3 ) (1,1,4) ( 5, 1, 4 ) ( 1, 1, 9 ) ( 1, 5, 4 ) ( 3, 1, 9 ) ( 3, 5, 4 ) (1,1,5) (5, 1, 5) ( 1, 1, 10) (1, 5, 5 ) (3, 1, 10 ) ( 3, 5, 5 ) (1,2,1) ( 3, 6, 6 ) ( 7, 2, 1 ) ( 1, 10, 1 ) ( 1, 6, 6 ) ( 3, 2, 6 ) ( 3, 6, 1 ) (1,2,2) ( 3, 6, 7 ) ( 5, 2, 2) ( 1, 6, 2 ) ( 1, 2, 7 ) ( 1, 6, 7 ) ( 3, 2, 7 ) ( 3, 6, 2 ) (1,2,3) ( 5, 2, 3 ) ( 1, 2, 8 ) ( 1, 6, 3 ) ( 3, 2, 8 ) ( 3, 6, 3 )

Realization [ 5, 7, 2, 13, 3, 8, 4, 14, 12, 1, 6, 10, 0, 11, 9 ] [ 1, 9, 7, 3, 5, 0, 2, 4, 6, 8 ] [ 5, 0, 10, 1, 6, 11, 2, 7, 9, 4, 13, 8, 3, 12 ] [ 1, 6, 2, 10, 5, 0, 9, 11, 7, 12, 3, 8, 4 ] [ 9, 0, 5, 3, 10, 4, 6, 1, 7, 2, 8 ] [ 0, 6, 2, 8, 4, 9, 1, 3, 7, 5 ] [ 11, 9, 5, 10, 6, 1, 15, 13, 8, 3, 14, 2, 7, 12, 0, 4 ] [ 10, 8, 6, 4, 2, 7, 9, 0, 5, 1, 3 ] [ 7, 3, 12, 8, 0, 4, 9, 11, 2, 6, 10, 1, 5 ] [ 6, 11, 1, 10, 0, 5, 7, 2, 12, 8, 3, 13, 4, 9 ] [ 11, 4, 6, 1, 9, 2, 7, 0, 5, 3, 10, 8 ] [ 10, 1, 6, 0, 7, 3, 5, 9, 2, 4, 8 ] [ 11, 15, 10, 6, 4, 16, 3, 8, 13, 1, 14, 9, 5, 7, 12, 0, 2 ] [ 1, 9, 4, 0, 2, 7, 5, 3, 8, 6 ] [ 8, 3, 7, 2, 6, 0, 4, 9, 1, 5 ] [ 2, 7, 1, 6, 0, 5, 9, 4, 10, 8, 3 ] [ 6, 11, 0, 10, 5, 1, 3, 13, 2, 7, 12, 8, 4, 14, 9 ] [ 12, 4, 6, 1, 10, 2, 7, 5, 0, 8, 3, 11, 9 ] [ 3, 7, 11, 1, 6, 2, 0, 8, 10, 5, 9, 4 ] [ 4, 6, 1, 10, 3, 5, 0, 9, 7, 2, 8 ] [ 6, 11, 1, 8, 3, 10, 5, 0, 4, 9, 2, 7 ] [ 1, 6, 0, 5, 9, 2, 7, 3, 10, 8, 4 ] [ 7, 12, 3, 8, 13, 1, 10, 5, 0, 2, 6, 11, 9, 4 ] [ 6, 11, 2, 7, 3, 1, 5, 9, 4, 0, 8, 10, 12 ] [ 9, 2, 7, 5, 0, 10, 8, 3, 11, 1, 6, 4 ] [ 4, 9, 1, 6, 11, 3, 5, 0, 8, 12, 7, 2, 10 ] [ 1, 5, 0, 8, 10, 6, 11, 3, 7, 2, 9, 4 ], [ 4, 5 ] [ 0, 5, 10, 12, 7, 2, 13, 8, 3, 1, 11, 6, 4, 14, 9 ] [ 6, 8, 3, 1, 11, 7, 2, 12, 10, 5, 0, 4, 9, 13 ] [ 5, 0, 12, 14, 9, 4, 2, 6, 1, 13, 8, 10, 15, 11, 7, 3 ] [ 1, 10, 8, 6, 2, 4, 0, 9, 3, 5, 7 ] [ 0, 4, 9, 5, 1, 10, 8, 12, 3, 7, 11, 2, 6 ] [ 10, 6, 1, 11, 7, 3, 12, 2, 0, 5, 9, 4, 13, 8 ] [ 10, 5, 0, 8, 1, 3, 7, 2, 4, 9, 11, 6 ] [ 6, 10, 3, 7, 9, 0, 4, 2, 8, 1, 5 ] [ 9, 14, 12, 0, 5, 1, 13, 11, 16, 3, 7, 2, 15, 10, 6, 8, 4 ] [ 9, 4, 6, 8, 0, 2, 7, 1, 5, 3 ] [ 2, 6, 0, 5, 9, 1, 7, 3, 8, 4 ] [ 4, 9, 3, 7, 1, 6, 2, 8, 10, 5, 0 ] [ 5, 0, 11, 7, 2, 12, 8, 3, 1, 6, 10, 14, 4, 9, 13 ] [ 1, 6, 11, 0, 9, 4, 2, 10, 5, 7, 12, 8, 3 ] [ 6, 11, 3, 7, 9, 1, 5, 0, 8, 10, 2, 4 ] [ 7, 9, 3, 5, 0, 2, 6, 1, 10, 8, 4 ] [ 3, 7, 2, 9, 4, 11, 6, 1, 8, 0, 10, 5 ] [ 4, 8, 1, 6, 10, 3, 7, 2, 9, 0, 5 ] [ 6, 11, 13, 4, 8, 3, 1, 10, 5, 0, 9, 7, 2, 12 ] [ 7, 11, 0, 8, 3, 12, 1, 5, 9, 4, 2, 6, 10 ]

.m

( 10, 4, 5 ) ( 7,–,–) (–,–, 4) (–, 3, 4) (8, –, 4 ) (2, 5,–) ( 13, 3, 9 ) ( 7,–,–) ( –, 3, –) (–,5, 6 ) ( 8,–, 4) (2, 6,–) ( 4, 8, 11) ( 6,–,–) (–, 3,–) (–,–,4) (–, 7, 9) ( 9,–, 4) (1, 4,–) (8, 3, 4) (–, 5, 6) ( –, 3, 4 ) ( –, 5, 7 ) ( –, 3, 6) ( 9, 3, 4 ) (–3, 4) (–,3, 4) (–, 5, 9) (–, 5, 6) (1, 10, 4) (6,–,–) (–, 3, –) (–, 6, 8) ( 1,–, 6) ( 7, 3, –) ( 10, 3, 4) (1,–,–) (–,5,–) (–,–, 4) (–, 9, 4) (2,–, 7) (7, 3,–) ( 8, 3, 4) (–,4, 5) (–, 3, 4 ) (–, 5, 6 ) (–, 4, 7 )

Method 5

5

5

4

4

5

5

4

The Buratti-Horak-Rosa Conjecture Holds for Some Underlying Sets of Size Three

415

Table 2 Growable realizations for .{2a , 4b , 5c } separated by congruence modulo .(2, 4, 5): case to .(1, 4, 4). They are x-growable at .mx in .m = (m2 , m4 , m5 )

.(1, 2, 4)

Case (a,b,c) (1,2,4) (5, 2, 4 ) ( 1, 2, 9 ) ( 1, 6, 4 ) ( 3, 2, 9 ) ( 3, 6, 4 ) (1,2,5) ( 3, 2, 5 ) ( 1, 2, 10 ) ( 1, 6, 5 ) (1,3,1) ( 3, 7, 6 ) ( 5, 3, 1 ) ( 1, 7, 1 ) ( 1, 3, 6 ) ( 1, 7, 6 ) ( 3, 3, 6 ) ( 3, 7, 1 ) (1,3,2) ( 3, 7, 7 ) ( 5, 3, 2 ) ( 1, 7, 2 ) ( 1, 3, 7) ( 1, 7, 7 ) ( 3, 3, 7 ) ( 3, 7, 2 ) (1,3,3) ( 1, 11, 3 ) ( 1, 3, 8 ) (5, 3, 3 ) ( 1, 7, 3 ) (1,3,4) ( 3, 3, 4 ) ( 1, 3, 9 ) ( 1, 7, 4 ) (1,3,5) (3, 3, 5 ) (1, 3, 10 ) ( 1, 7, 5 ) (1,4,1) ( 1, 4, 11 ) (1, 8, 1 ) ( 5, 4, 1 ) ( 1, 4, 6 ) (1,4,2) (1, 8, 2 ) ( 1, 4, 7 ) ( 5, 4, 2 ) (1,4,3) ( 3, 4, 3 ) ( 1, 4, 8 ) ( 1, 8, 3) (1,4,4) (1, 4, 9 ) ( 1, 4, 4 )

Realization [ 8, 10, 3, 7, 2, 0, 5, 9, 11, 1, 6, 4 ] [ 1, 6, 11, 2, 10, 5, 3, 8, 0, 9, 4, 12, 7 ] [ 6, 10, 3, 8, 0, 4, 9, 5, 1, 11, 7, 2 ] [ 8, 13, 3, 14, 12, 7, 2, 4, 9, 11, 1, 6, 10, 5, 0 ] [ 0, 12, 2, 7, 11, 1, 10, 6, 4, 9, 5, 3, 13, 8 ] [ 5, 0, 7, 2, 9, 3, 8, 10, 1, 6, 4 ] [ 13, 9, 0, 5, 10, 1, 6, 11, 2, 4, 8, 3, 12, 7 ] [ 6, 10, 1, 5, 7, 2, 11, 3, 8, 12, 4, 9, 0 ] [ 10, 15, 11, 6, 2, 7, 12, 8, 4, 0, 13, 1, 3, 5, 9, 14, 16 ] [ 2, 6, 4, 8, 0, 5, 7, 3, 1, 9 ] [ 5, 1, 7, 3, 9, 4, 0, 8, 2, 6 ] [ 1, 6, 10, 4, 9, 5, 3, 8, 2, 7, 0 ] [ 8, 12, 7, 3, 14, 4, 9, 5, 0, 10, 6, 2, 13, 11, 1 ] [ 5, 0, 8, 6, 1, 10, 12, 4, 9, 11, 2, 7, 3 ] [ 9, 7, 3, 11, 1, 5, 0, 8, 4, 6, 2, 10 ] [ 11, 6, 2, 16, 0, 5, 9, 4, 17, 3, 7, 12, 10, 14, 1, 15, 13, 8 ] [ 1, 3, 8, 10, 5, 7, 0, 9, 2, 6, 4 ] [ 1, 10, 6, 2, 8, 4, 0, 7, 3, 9, 5 ] [ 10, 5, 0, 8, 3, 1, 6, 2, 9, 4, 11, 7 ] [ 0, 4, 8, 3, 5, 10, 15, 11, 6, 2, 14, 9, 13, 1, 12, 7 ] [ 4, 6, 1, 11, 2, 7, 5, 0, 10, 12, 3, 8, 13, 9 ] [ 10, 12, 1, 6, 2, 11, 3, 7, 5, 9, 0, 4, 8 ] [ 7, 3, 15, 11, 0, 5, 9, 13, 1, 6, 2, 14, 10, 12, 8, 4 ] [ 10, 2, 6, 1, 9, 11, 3, 7, 12, 4, 8, 0, 5 ] [ 5, 10, 0, 2, 7, 3, 1, 9, 11, 6, 4, 8 ] [ 7, 3, 11, 4, 8, 10, 2, 6, 1, 9, 5, 0 ] [ 4, 8, 10, 3, 5, 0, 9, 2, 7, 1, 6 ] [ 5, 7, 2, 12, 3, 8, 13, 4, 9, 0, 10, 1, 11, 6 ] [ 7, 12, 4, 8, 3, 11, 2, 0, 9, 5, 1, 10, 6 ] [ 7, 2, 10, 3, 8, 0, 5, 9, 11, 1, 6, 4 ] [ 0, 10, 5, 1, 11, 6, 4, 9, 14, 3, 8, 13, 2, 12, 7 ] [ 4, 9, 13, 8, 10, 0, 5, 1, 11, 6, 2, 12, 3, 7 ] [ 5, 9, 4, 16, 11, 7, 2, 14, 12, 0, 13, 1, 6, 10, 15, 3, 8 ] [ 6, 2, 9, 0, 7, 3, 10, 4, 8, 1, 5 ] [ 4, 6, 8, 1, 10, 3, 7, 2, 0, 9, 5 ] [ 5, 10, 2, 4, 9, 1, 6, 11, 7, 0, 8, 3 ] [ 5, 9, 1, 6, 2, 10, 0, 8, 4, 11, 3, 7 ] [ 5, 0, 9, 4, 12, 8, 6, 1, 10, 2, 7, 3, 11 ] [ 4, 8, 3, 5, 7, 11, 9, 1, 6, 2, 0, 10 ] [ 4, 8, 2, 6, 1, 10, 3, 7, 9, 0, 5 ] [ 7, 11, 1, 6, 10, 5, 0, 9, 4, 2, 12, 3, 8, 13 ] [ 6, 10, 5, 1, 9, 11, 2, 7, 3, 12, 8, 4, 0 ] [ 4, 6, 1, 11, 7, 2, 12, 8, 3, 13, 9, 14, 10, 0, 5 ] [ 5, 1, 6, 2, 7, 3, 8, 0, 4, 9 ]

.m

( 9, 3, 4 ) (–, 6, 7) ( –, 3, 6) (–, 7, 8) ( –,7, 8) (8, 3, 4) (–, 3, 7 ) ( –, 4, 6) ( 1, 7, 10 ) (–,–, 2) (–,3,–) (–,–, 5) (–, 5, 8) ( 9,–, 4 ) ( 8, 3,– ) (14, 5, 8) ( 2,–,–) (–,5,–) ( –,–,4) (–, 5, 7) (9,–, 4) (10, 3,–) (9, 3, 4) ( 8, –, 4 ) ( 1,–, 6 ) ( 9,–, 4 ) ( 8, 3, 4 ) (–, 5, 6) (–, 6, 7 ) ( 9, 3, 4 ) (–,6, 7) ( –, 3, 4 ) ( 11, 4, 5 ) ( 8, 4, –) ( 3, 5, – ) ( 2, 6, –) ( 9, 4, 5 ) ( 9,–, 4) ( 1,–, 5) ( 8, 3, 4) (–,3, 7) (–,3, 6) ( 8, 3, 4 ) ( 1, 3, –)

Method 4

3

5

5

4

3

3

4

3

3

2

416

P. Chand and M. A. Ollis

Table 3 Growable realizations for .{2a , 4b , 5c } separated by congruence modulo .(2, 4, 5): case to .(2, 3, 2). They are x-growable at .mx in .m = (m2 , m4 , m5 )

.(1, 4, 5)

Case (a,b,c) (1,4,5) ( 1, 4, 10 ) ( 1, 4, 5 ) (2,1,1) ( 4, 5, 6 ) ( 8, 1, 1) ( 2, 9, 1 ) ( 2, 5, 6 ) ( 4, 1, 6 ) ( 4, 5, 1 ) (2,1,2) ( 2, 1, 12) ( 2, 5, 2 ) (8, 1, 2 ) (2, 5, 7 ) (4, 1, 7 ) (2,1,3) (2, 5, 3 ) (2, 1, 8 ) ( 6, 1, 3 ) (2,1,4) (2, 5, 4 ) ( 2, 1, 9 ) ( 6, 1, 4 ) (2,1,5) (2, 5, 5 ) ( 2, 1, 10 ) ( 4, 1, 5 ) (2,2,1) (2, 2, 11) ( 2, 6, 1 ) ( 6, 2, 1 ) (2, 2, 6 ) (2,2,2) ( 2, 6, 2) ( 2, 2, 7 ) ( 8, 2, 2 ) (2,2,3) (2, 6, 3 ) ( 2, 2, 8 ) ( 6, 2, 3 ) (2,2,4) (2, 6, 4 ) ( 2, 2, 9 ) ( 4, 2, 4 ) (2,2,5) ( 2, 2, 10 ) ( 2, 6, 5 ) ( 4, 2, 5) (2,3,1) ( 2, 3, 11 ) ( 2, 7, 1 ) ( 6, 3, 1 ) ( 2, 3, 6 ) (2,3,2) ( 2, 3, 12 ) ( 2, 11, 2 ) ( 4, 3, 2 ) ( 2, 3, 7 )

Realization [ 7, 2, 13, 9, 4, 15, 11, 0, 12, 1, 6, 8, 3, 14, 10, 5 ] [ 3, 7, 2, 8, 1, 5, 10, 6, 0, 9, 4 ] [ 9, 13, 1, 12, 7, 2, 14, 0, 4, 15, 3, 5, 10, 8, 6, 11 ] [ 10, 1, 8, 6, 4, 2, 0, 9, 3, 5, 7 ] [ 7, 3, 12, 8, 4, 0, 2, 6, 11, 9, 5, 1, 10 ] [ 0, 5, 1, 10, 12, 8, 3, 13, 9, 4, 2, 7, 11, 6 ] [ 3, 7, 2, 9, 4, 11, 1, 6, 8, 10, 0, 5 ] [ 3, 7, 0, 2, 6, 8, 10, 1, 5, 9, 4 ] [ 4, 9, 14, 0, 11, 15, 10, 5, 3, 8, 13, 2, 7, 12, 1, 6 ] [ 5, 1, 7, 3, 8, 6, 2, 0, 4, 9 ] [ 7, 9, 11, 1, 6, 8, 4, 2, 0, 10, 5, 3 ] [ 7, 9, 14, 3, 13, 8, 4, 2, 12, 1, 5, 10, 0, 11, 6 ] [ 6, 8, 3, 11, 0, 2, 10, 1, 9, 4, 12, 7, 5 ] [ 4, 8, 10, 3, 7, 1, 6, 2, 9, 0, 5 ] [ 11, 4, 6, 1, 8, 3, 10, 0, 5, 9, 2, 7 ] [ 3, 8, 6, 4, 2, 7, 9, 0, 5, 1, 10 ] [ 6, 1, 9, 11, 7, 2, 10, 3, 5, 0, 8, 4 ] [ 3, 8, 12, 7, 2, 4, 9, 1, 6, 11, 0, 5, 10 ] [ 11, 1, 6, 2, 4, 9, 7, 5, 0, 10, 8, 3 ] [ 4, 8, 3, 12, 10, 6, 1, 9, 5, 0, 11, 2, 7 ] [ 5, 10, 1, 6, 11, 2, 4, 9, 0, 12, 7, 3, 8, 13 ] [ 8, 2, 7, 9, 3, 5, 0, 4, 6, 1, 10 ] [ 7, 2, 13, 8, 3, 5, 10, 15, 11, 6, 1, 12, 0, 14, 9, 4 ] [ 6, 2, 0, 8, 4, 9, 5, 1, 7, 3 ] [ 5, 7, 3, 1, 9, 4, 6, 2, 0, 8 ] [ 2, 7, 1, 3, 8, 10, 6, 0, 5, 9, 4 ] [ 6, 2, 9, 3, 7, 0, 5, 1, 10, 8, 4 ] [ 4, 9, 1, 3, 8, 10, 5, 0, 7, 2, 6, 11 ] [ 9, 11, 0, 5, 1, 12, 10, 8, 6, 4, 2, 7, 3 ] [ 6, 1, 8, 4, 0, 10, 2, 7, 3, 11, 9, 5 ] [ 7, 2, 11, 3, 8, 6, 1, 9, 4, 0, 5, 10, 12 ] [ 3, 5, 0, 10, 6, 1, 11, 9, 7, 2, 4, 8 ] [ 8, 3, 12, 4, 9, 0, 11, 2, 7, 5, 1, 10, 6 ] [ 3, 8, 12, 7, 2, 4, 9, 13, 11, 6, 1, 10, 5, 0 ] [ 3, 8, 10, 1, 5, 0, 6, 2, 4, 9, 7 ] [ 6, 11, 1, 12, 2, 7, 9, 4, 14, 10, 5, 0, 13, 3, 8 ] [ 12, 0, 5, 1, 10, 6, 8, 4, 9, 13, 3, 7, 2, 11 ] [ 0, 10, 3, 8, 1, 5, 7, 9, 2, 6, 4, 11 ] [ 4, 8, 3, 15, 10, 5, 0, 13, 1, 6, 11, 16, 12, 14, 2, 7, 9 ] [ 5, 9, 2, 4, 8, 1, 10, 3, 7, 0, 6 ] [ 10, 1, 6, 8, 4, 2, 0, 9, 5, 3, 7 ] [ 11, 4, 6, 10, 3, 5, 1, 8, 0, 7, 2, 9 ] [ 3, 8, 13, 17, 12, 7, 2, 4, 9, 14, 10, 15, 1, 6, 11, 16, 0, 5 ] [ 6, 2, 13, 1, 5, 9, 7, 11, 15, 3, 8, 4, 0, 12, 14, 10 ] [ 0, 5, 7, 3, 1, 9, 4, 8, 6, 2 ] [ 6, 1, 9, 0, 8, 4, 12, 10, 2, 7, 11, 3, 5 ]

.m

(10, 4, 5 ) ( 2, 4,–) (13, 8, 9 ) (3,–,–) (–, 3,–) ( –, 9, 4 ) (9,–, 4 ) ( 1, 4,–) ( 2, 10, 5 ) ( 7, 3,–) ( 1, 7,–) ( 5, 7,–) ( 4, 6,–) (8, 3, 4 ) (9,–, 4) ( 1,–, 4) (8, 3, 4) (1,–, 4) (1,–, 4) (10, 3, 4) ( 2,–, 7) ( 8,–, 4) ( 2, 11, 5) (1, 4,–) (1, 5,–) ( 1, 5,–) ( 8, 3, 4) ( 1,–, 6) (1,– 4) (9, 3, 5 ) ( 10,–, 4) ( 1,–, 4) ( 10, 4, 6 ) ( 1,–, 4) (1,–, 4 ) ( 11, 5, 6) ( 10, 3,–) ( 8, 4, –) (11, 3, 4) ( 3, 6,–) (1, 6,–) (7, 4,–) ( 1, 9, 4) ( 11, 3,–) (2, 5,–) (8, 4,–)

Method 2 5

4

3

4

3

4

3

3

3

3

4

4

The Buratti-Horak-Rosa Conjecture Holds for Some Underlying Sets of Size Three

417

Table 4 Growable realizations for .{2a , 4b , 5c } separated by congruence modulo .(2, 4, 5): case to .(2, 4, 5). They are x-growable at .mx in .m = (m2 , m4 , m5 )

.(2, 3, 3)

Case (a,b,c) (2,3,3) ( 2, 3, 8 ) ( 2, 7, 3 ) ( 4, 3, 3 ) (2,3,4) (2, 3, 9 ) (2, 3, 4 ) (2,3,5) (2, 3, 10) ( 2, 3, 5 ) (2,4,1) ( 2, 4, 11 ) ( 2, 8, 1 ) ( 4, 4, 1 ) ( 2, 4, 6 ) (2,4,2) (2, 4, 7) ( 2, 8, 2 ) ( 4, 4, 2 ) (2,4,3) (4, 4, 3) (2, 4, 3 ) (2,4,4) ( 2, 4, 9 ) (2, 4, 4 ) (2,4,5) (2, 4, 5 )

Realization [ 6, 10, 1, 11, 13, 4, 9, 0, 5, 7, 2, 12, 3, 8 ] [ 6, 8, 4, 12, 3, 7, 11, 2, 0, 9, 1, 10, 5 ] [ 1, 6, 10, 8, 3, 5, 9, 4, 2, 0, 7 ] [ 7, 2, 12, 14, 10, 5, 0, 11, 1, 6, 8, 3, 13, 9, 4 ] [ 1, 6, 2, 0, 5, 7, 3, 8, 4, 9 ] [ 6, 11, 15, 4, 9, 7, 2, 13, 1, 12, 8, 3, 14, 0, 5, 10 ] [ 6, 0, 2, 8, 4, 10, 1, 7, 3, 9, 5 ] [ 6, 11, 16, 12, 7, 2, 15, 13, 0, 14, 1, 5, 10, 8, 3, 17, 4, 9 ] [ 3, 7, 11, 9, 1, 5, 10, 6, 2, 0, 8, 4 ] [ 5, 7, 3, 9, 1, 6, 2, 0, 8, 4 ] [ 12, 4, 6, 2, 10, 1, 9, 0, 8, 3, 11, 7, 5 ] [ 4, 8, 13, 3, 7, 12, 10, 1, 6, 2, 11, 9, 0, 5 ] [ 11, 3, 7, 5, 9, 0, 4, 8, 12, 10, 1, 6, 2 ] [ 8, 10, 3, 7, 1, 5, 0, 9, 2, 4, 6 ] [ 11, 1, 3, 7, 2, 4, 9, 5, 0, 8, 10, 6 ] [ 3, 7, 2, 6, 1, 9, 5, 0, 8, 4 ] [ 10, 5, 0, 4, 8, 13, 15, 3, 14, 2, 7, 12, 1, 6, 11, 9 ] [ 9, 3, 5, 10, 6, 1, 8, 2, 4, 0, 7 ] [ 5, 0, 8, 3, 10, 2, 7, 11, 9, 1, 6, 4 ]

.m

(10, 5, 6) (4, 6,–) (1, 6,–) (10, 3, 4) (3, 5,–) (12, 5, 6) (4, 6,–) ( 12, 4, 6) ( 1, 4,–) ( 3, 5,–) ( 8, 5,–) (8, 3, 4) (9, 3,–) (8, 3,–) (1, 5, 6) ( 3, 5,–) (13, 7, 9 ) (6, 3,–) ( 8, 3, 4)

Method 3

2 2 4

3

2 2 1

most .2v − (μ − 1)v/μ = v − v/μ of these are not with edges of length x. Hence there are insufficiently many non-x incidences to avoid having a run of .μ vertices each incident with two edges of length x. ⨆ ⨅ The following result allows Lemma 11 to be useful for the underlying sets of interest in this section. Theorem 12 ([8]) Let .x ≥ 4 be even. The multiset .{1a , 2b , x c } satisfies the BHR Conjecture if .a + b ≥ x − 1. We are therefore concerned here with multisets .{1a , 2b , x c } that have even .x ≥ 4 and with .a + b < x − 1. We start by putting an upper bound on the smallest counterexample. Theorem 13 Let .x ≥ 4 be even and .L = {1a , 2b , x c }. If there is a counterexample to the BHR Conjecture for a multiset of this form, then there is one with .a+b < x−1 and .c < x 2 − x + 1. Proof First, by Theorem 12, every counterexample must have .a + b < x − 1. We apply Lemma 11: Any realization with underlying set U and .c ≥ xv/(x − 1) must be x-growable. As .v = a + b + c + 1 and .a + b ≤ x − 2, this simplifies to .c ≥ x 2 − 2x + 1. Therefore, if the BHR Conjecture holds for all c in the range 2 2 .x − 2x + 1 ≤ c < x − x + 1 it holds for all greater values of c too, by the x-

418

P. Chand and M. A. Ollis

Table 5 Growable realizations for .{3a , 4b , 5c } separated by congruence modulo .(3, 4, 5): case to .(1, 3, 2). They are x-growable at .mx in .m = (m3 , m4 , m5 )

.(1, 1, 1)

Case (a,b,c) (1,1,1) (4,5,6) (1, 1, 11) ( 7, 1, 1 ) ( 1, 5, 6 ) ( 4, 1, 6 ) ( 4, 5, 1 ) (1,1,2) (1, 9, 2 ) (7, 1, 2 ) (4, 1, 7 ) (1, 5, 7 ) (1,1,3) ( 10, 1, 3 ) (1, 1, 8 ) ( 1, 9, 3 ) ( 4, 1, 8 ) ( 4, 5, 3 ) ( 7, 1, 3 ) (1,1,4) ( 7, 1, 4 ) ( 1, 1, 9 ) ( 1, 5, 4 ) ( 4, 1, 9 ) ( 4, 5, 4 ) (1,1,5) ( 4, 1, 5 ) ( 1, 1, 10 ) ( 1, 5, 5 ) (1,2,1) ( 4, 6, 6 ) ( 7, 2, 1 ) ( 1, 10, 1) ( 1, 6, 6 ) ( 4, 2, 6 ) ( 4, 6, 1 ) (1,2,2) ( 1, 2, 7 ) ( 1, 6, 2 ) ( 7, 2, 2 ) (1,2,3) ( 1, 2, 8 ) ( 1, 6, 3 ) ( 4, 2, 3 ) (1,2,4) ( 1, 2, 9 ) ( 1, 6, 4 ) ( 4, 2, 4) (1,2,5) ( 4, 2, 5 ) ( 1, 2, 10 ) ( 1, 6, 5 )

Realization [ 5, 10, 14, 2, 7, 4, 15, 11, 8, 3, 0, 12, 1, 6, 9, 13 ] [ 12, 7, 2, 11, 6, 1, 10, 5, 0, 3, 8, 13, 9, 4 ] [ 2, 5, 9, 6, 3, 8, 1, 4, 7, 0 ] [ 6, 11, 7, 2, 10, 1, 5, 9, 4, 12, 8, 3, 0 ] [ 9, 0, 5, 2, 10, 7, 4, 11, 6, 1, 8, 3 ] [ 5, 9, 6, 2, 10, 3, 8, 1, 4, 7, 0 ] [ 5, 9, 1, 10, 6, 2, 11, 7, 3, 0, 4, 12, 8 ] [ 4, 7, 10, 5, 2, 9, 1, 6, 3, 0, 8 ] [ 4, 9, 1, 6, 3, 12, 7, 2, 10, 0, 5, 8, 11 ] [ 4, 8, 13, 9, 0, 5, 2, 12, 3, 7, 11, 6, 1, 10 ] [ 6, 9, 12, 0, 3, 8, 11, 14, 2, 7, 4, 1, 5, 10, 13 ] [ 5, 10, 4, 9, 1, 6, 0, 7, 2, 8, 3 ] [ 6, 10, 13, 3, 7, 2, 11, 1, 5, 9, 0, 4, 8, 12 ] [ 12, 3, 6, 11, 1, 4, 9, 0, 5, 10, 7, 2, 13, 8 ] [ 3, 8, 4, 0, 10, 5, 1, 11, 2, 6, 9, 12, 7 ] [ 10, 1, 4, 9, 0, 3, 7, 2, 5, 8, 11, 6 ] [ 11, 1, 6, 9, 12, 8, 3, 0, 5, 2, 10, 7, 4 ] [ 6, 11, 7, 2, 9, 4, 1, 8, 3, 10, 5, 0 ] [ 6, 1, 8, 0, 4, 9, 2, 7, 3, 10, 5 ] [ 4, 9, 12, 7, 2, 14, 10, 5, 0, 3, 8, 13, 1, 11, 6 ] [ 6, 10, 5, 1, 4, 9, 13, 2, 7, 11, 0, 3, 12, 8 ] [ 5, 10, 7, 2, 8, 0, 4, 9, 1, 6, 3 ] [ 5, 0, 10, 2, 7, 12, 8, 3, 11, 6, 1, 9, 4 ] [ 5, 9, 4, 1, 6, 2, 10, 3, 8, 0, 7, 11 ] [ 5, 10, 6, 2, 15, 1, 14, 11, 16, 4, 9, 12, 7, 3, 0, 13, 8 ] [ 7, 10, 2, 8, 5, 1, 4, 0, 3, 6, 9 ] [ 4, 8, 12, 7, 3, 0, 9, 5, 1, 10, 6, 2, 11 ] [ 6, 10, 0, 5, 9, 13, 4, 8, 3, 12, 1, 11, 2, 7 ] [ 5, 8, 3, 12, 4, 7, 2, 10, 0, 9, 1, 11, 6 ] [ 10, 6, 3, 0, 4, 8, 5, 1, 9, 2, 11, 7 ] [ 4, 9, 2, 7, 1, 6, 10, 5, 0, 8, 3 ] [ 8, 2, 6, 1, 5, 9, 4, 0, 7, 3 ] [ 5, 2, 11, 3, 8, 0, 9, 6, 1, 10, 7, 4 ] [ 5, 10, 1, 6, 11, 3, 8, 0, 7, 2, 9, 4 ] [ 8, 1, 5, 9, 4, 0, 6, 10, 2, 7, 3 ] [ 8, 5, 0, 3, 7, 4, 9, 2, 6, 1 ] [ 4, 9, 12, 7, 2, 11, 6, 1, 10, 5, 0, 8, 3 ] [ 4, 8, 0, 5, 2, 9, 1, 6, 10, 3, 7, 11 ] [ 3, 8, 0, 5, 9, 1, 6, 10, 2, 7, 4 ] [ 6, 9, 4, 11, 2, 7, 3, 0, 8, 1, 10, 5 ] [ 8, 11, 2, 7, 12, 3, 13, 4, 9, 0, 5, 10, 6, 1 ] [ 12, 3, 8, 4, 0, 5, 9, 1, 11, 7, 2, 10, 6 ]

.m

( 10, 4, 5 ) (–, –, 4 ) (3,–,–) (–, 4, 6 ) (8,–, 4) (3, 5,–) (7, 8,–) (2, 4,– ) (9, 3, 4 ) ( 8, 3, 4 ) (10, 4, 6 ) (–, 4, 5 ) ( –, 3, 6) (–, 3, 8) (–, 3, 7 ) (–, 5, 6) ( 8, 3, 4 ) (–, 3, 6) (–, 3, 5) (–,5, 6) (–, 7, 8 ) (2, 3, 5) (–, 3, 4 ) (–,4, 5) ( 4, 5, 8 ) ( 7,–,–) (–, 4,–) (–, 5, 6) (4,–, 6 ) (5, 7,– ) ( 2, 3, 4 ) ( 2, 3, – ) (3, 4, –) ( 3, 4, 5 ) ( 2, 3 ,– ) ( 4, 5,–) ( 2, 3, 4 ) ( 8, 3, –) (2, 3, –) ( 4, 5, 6) ( –, 7, 8) (–, 5, 6)

Method 5

3

4

4

3

5

3

3

3

3

(continued)

The Buratti-Horak-Rosa Conjecture Holds for Some Underlying Sets of Size Three

419

Table 5 (continued) Case (1,3,1)

(1,3,2)

(a,b,c) ( 1, 3, 6) ( 1, 7, 1) ( 7, 3, 1) ( 1, 3, 7 ) ( 1, 7, 2 ) ( 7, 1, 2)

Realization [ 5, 10, 6, 1, 7, 2, 9, 4, 0, 8, 3 ] [ 5, 9, 4, 8, 2, 6, 0, 3, 7, 1 ] [ 4, 7, 10, 1, 5, 9, 0, 3, 8, 11, 2, 6 ] [ 5, 10, 2, 7, 0, 4, 9, 1, 6, 11, 8, 3 ] [ 4, 9, 2, 6, 10, 5, 1, 8, 0, 7, 3 ] [ 3, 6, 9, 1, 4, 0, 8, 5, 10, 2, 7 ]

.m

(2, 3, 5) (4, 5, –) ( 2, 4, –) ( 2, 3, 5 ) ( 3, 4,–) (2, 3,–)

Method 3

3

growability of the realizations for these values. Hence if there is a counterexample, then there must be one with .c < x 2 − x + 1, as required. ⨆ ⨅ Therefore, to prove .BHR(L) for .U = {1, 2, x} with x even, it suffices to show that there is no counterexample with .v ≤ (x − 2) + (x 2 − x) + 1 = x 2 − 1, a finite process. In any given case, we expect to achieve the result by checking many fewer cases than potentially required by this upper bound. Indeed, in performing the computations for the following result, we did not encounter a situation where a multiset is admissible but an x-growable realization does not exist. Theorem 14 Let .4 ≤ x ≤ 12 with x even. .BHR(L) holds for multisets L with underlying set .{1, 2, x}. Proof The result is proved for .x ∈ {4, 6, 8} in [8]. For each of the remaining cases we do the following. For each pair .a, b with a b k .a + b < x − 1, let k be the smallest value such that .{1 , 2 , x } is admissible. We a b c find x-growable realizations for .{1 , 2 , x } for .k ≤ c ≤ k + x − 1, excluding values of c such that .{1a , 2b , x c+ix } is inadmissible for all .i ≥ 0. This covers all admissible multisets. The required realizations are given in a text file on the ArXiv page for this paper, or are available on request from the second author. There are 249 for the .x = 10 case and 487 for the .x = 12 case. ⨆ ⨅

5 An Algorithm The third and final approach to underlying sets of size 3 has much in common with that of Sect. 3. As in that section, let .U = {x, y, z} and .L = {x a , y b , zc }. We again divide the work for a given U into .π(U ) cases according to the combinations of congruences of a, b and c modulo x, y and z respectively. Suppose we are working on one case and have covered all .(a, b, c) with .a < a0 , .b < b0 or .c < c0 for some .(a0 , b0 , c0 ) in the appropriate equivalence class modulo .(x, y, z). As in Sect. 3, define .ai to be .a0 + ix and make similar definitions for .bi and .ci with respect to y and z respectively. Consider the following steps:

420

P. Chand and M. A. Ollis

Table 6 Growable realizations for .{3a , 4b , 5c } separated by congruence modulo .(3, 4, 5): case to .(2, 2, 1). They are x-growable at .mx in .m = (m3 , m4 , m5 )

.(1, 3, 3)

Case (a,b,c) (1,3,3) ( 1, 3, 8 ) ( 1, 7, 3 ) ( 4, 3, 3 ) (1,3,4) ( 4, 3, 4 ) ( 1, 3, 9 ) ( 1, 7, 4 ) (1,3,5) (1, 3, 10 ) (1, 7, 5 ) ( 4, 3, 5 ) (1,4,1) ( 1, 4, 11 ) ( 1, 8, 1 ) ( 4, 4, 1 ) ( 1, 4, 6 ) (1,4,2) ( 1, 4, 7 ) ( 1, 8, 2 ) ( 4, 4, 2 ) (1,4,3) ( 1, 4, 8 ) ( 1, 8, 3 ) ( 4, 4, 3 ) (1,4,4) ( 1, 4, 9 ) ( 1, 8, 4 ) ( 4, 4, 4 ) (1,4,5) ( 1, 4, 5 ) (2,1,1) ( 5, 5, 6 ) ( 8, 1, 1 ) ( 2, 9, 1 ) ( 2, 5, 6 ) ( 5, 1, 6 ) ( 5, 5, 1 ) (2,1,2) ( 2, 1, 12) ( 2, 5, 2 ) ( 8, 1, 2 ) ( 2, 1, 7 ) (2,1,3) ( 2, 5, 3 ) ( 2, 1, 13) ( 8, 1, 3 ) (2,1,4) ( 8, 1, 4 ) ( 2, 1, 9 ) ( 2, 5, 4 ) ( 5, 1, 4 ) (2,1,5) ( 2, 1, 10 ) ( 2, 5, 5 ) ( 5, 1, 5 ) (2,2,1) ( 2, 2, 6 ) ( 2, 6, 1) ( 8, 2, 1)

Realization [ 6, 11, 2, 7, 12, 3, 8, 0, 10, 5, 1, 9, 4 ] [ 9, 0, 4, 8, 1, 5, 10, 2, 6, 11, 7, 3 ] [ 4, 7, 0, 8, 3, 9, 1, 5, 10, 2, 6 ] [ 4, 7, 0, 5, 1, 9, 6, 2, 11, 8, 3, 10 ] [ 10, 5, 1, 6, 11, 2, 7, 12, 3, 0, 4, 9, 13, 8 ] [ 4, 8, 12, 7, 3, 11, 2, 6, 1, 10, 0, 5, 9 ] [ 7, 12, 2, 6, 11, 1, 13, 8, 3, 14, 9, 4, 0, 10, 5 ] [ 6, 11, 1, 5, 10, 0, 9, 12, 2, 7, 3, 13, 8, 4 ] [ 11, 2, 5, 0, 8, 3, 12, 9, 1, 6, 10, 7, 4 ] [ 10, 15, 11, 16, 3, 7, 12, 0, 5, 8, 13, 1, 6, 2, 14, 9, 4 ] [ 6, 0, 7, 3, 10, 2, 9, 5, 1, 8, 4 ] [ 6, 0, 4, 7, 3, 8, 1, 5, 2, 9 ] [ 4, 9, 1, 6, 11, 8, 0, 5, 10, 2, 7, 3 ] [ 5, 9, 1, 6, 2, 10, 0, 8, 4, 12, 7, 3, 11 ] [ 11, 3, 7, 10, 6, 2, 9, 1, 5, 0, 8, 4 ] [ 5, 8, 0, 7, 3, 10, 2, 6, 1, 9, 4 ] [ 4, 7, 12, 3, 8, 13, 9, 5, 0, 10, 1, 6, 2, 11 ] [ 5, 9, 0, 8, 4, 12, 3, 7, 2, 11, 1, 10, 6 ] [ 4, 8, 11, 3, 7, 0, 5, 2, 10, 1, 6, 9 ] [ 10, 0, 5, 9, 14, 4, 8, 13, 3, 6, 1, 12, 2, 7, 11 ] [ 7, 11, 1, 5, 10, 0, 4, 9, 6, 2, 12, 3, 13, 8 ] [ 4, 9, 0, 10, 6, 1, 11, 2, 5, 8, 3, 12, 7 ] [ 4, 9, 5, 0, 7, 2, 10, 6, 1, 8, 3 ] [ 7, 12, 8, 4, 16, 3, 0, 13, 9, 6, 1, 15, 10, 5, 2, 14, 11 ] [ 10, 2, 5, 8, 0, 4, 7, 1, 9, 6, 3 ] [ 9, 5, 1, 6, 10, 0, 4, 8, 12, 2, 11, 7, 3 ] [ 9, 0, 5, 2, 11, 1, 6, 10, 13, 4, 8, 12, 7, 3 ] [ 0, 5, 2, 10, 7, 4, 1, 6, 11, 3, 8, 12, 9 ] [ 1, 5, 8, 4, 0, 9, 2, 11, 3, 6, 10, 7 ] [ 5, 10, 15, 4, 7, 2, 13, 8, 3, 14, 9, 6, 1, 12, 0, 11 ] [ 0, 5, 9, 3, 6, 2, 7, 1, 4, 8 ] [ 9, 0, 3, 6, 2, 11, 4, 7, 10, 1, 8, 5 ] [ 2, 7, 1, 8, 0, 6, 3, 9, 4, 10, 5 ] [ 3, 8, 4, 0, 7, 2, 10, 6, 1, 9, 5 ] [ 15, 10, 5, 0, 12, 16, 11, 6, 3, 8, 13, 1, 4, 9, 14, 2, 7 ] [ 9, 12, 2, 5, 0, 3, 8, 11, 1, 6, 10, 7, 4 ] [ 6, 9, 13, 10, 5, 2, 7, 4, 1, 12, 3, 8, 11, 0 ] [ 0, 5, 8, 3, 11, 6, 1, 10, 2, 7, 12, 9, 4 ] [ 5, 9, 2, 6, 1, 10, 3, 8, 0, 4, 7, 11 ] [ 6, 1, 9, 4, 10, 2, 7, 3, 0, 8, 5 ] [ 5, 8, 13, 4, 9, 0, 10, 1, 6, 3, 12, 7, 2, 11 ] [ 3, 7, 10, 1, 9, 5, 0, 8, 4, 12, 2, 11, 6 ] [ 10, 1, 5, 8, 3, 0, 7, 2, 11, 6, 9, 4 ] [ 4, 8, 3, 9, 1, 6, 0, 7, 2, 10, 5 ] [ 3, 7, 1, 5, 8, 2, 6, 9, 4, 0 ] [ 3, 6, 9, 0, 4, 1, 8, 5, 2, 11, 7, 10 ]

.m

( 3, 4, 6 ) ( 2, 3,–) ( 3, 4,– ) ( 8, 3, 4 ) ( –, 7, 8 ) ( –, 3, 4 ) ( 4, 5, 7) ( 3, 4, – ) ( 3, 4, – ) ( 9, 3, 4 ) ( 5, 6, – ) ( 2, 3,– ) ( 2, 3,–) ( 9, 4, 5 ) (8, 3, – ) ( 3, 4, –) ( 9, 3, 4 ) (4, 5, – ) ( 8, 3, – ) ( 9, 10, 4) (6, 7, –) ( 3, 4, –) (2, 3, 4) (6, 7, 11) ( 4,–,–) (–, 6,–) ( –, 9, 4) ( 8,–, 4) ( 5, 7, –) (10, 4, 5) ( 3, 5, –) ( 4, 5,–) (4, 5,–) ( 2, 3, 5 ) ( 2,–, 6 ) ( 8,–, 4 ) (10, 5, 6 ) (–, 3, 4) (–,3, 5) (–, 4, 5) (9, 4, 5) ( 5, 6, – ) (2, 4, – ) ( 3, 4, 5 ) (2, 3, –) ( 2, 3, –)

Method 3

3

3

4

3

3

3

1 5

4

3

4

3

3

The Buratti-Horak-Rosa Conjecture Holds for Some Underlying Sets of Size Three

421

Table 7 Growable realizations for .{3a , 4b , 5c } separated by congruence modulo .(3, 4, 5): case to .(3, 1, 4). They are x-growable at .mx in .m = (m3 , m4 , m5 )

.(2, 2, 2)

Case (2,2,2)

(2,2,3)

(2,2,4)

(2,2,5)

(2,3,1)

(2,3,2)

(2,3,3)

(2,3,4) (2,3,5) (2,4,1)

(2,4,2)

(2,4,3) (2,4,4) (2,4,5) (3,1,1)

(3,1,2)

(3,1,3)

(3,1,4)

(a,b,c) ( 2, 2, 7) ( 2, 6, 2 ) ( 5, 2, 2 ) ( 2, 2, 8) ( 2, 6, 3) ( 5, 2, 3 ) ( 2, 2, 9 ) ( 2, 6, 4 ) ( 5, 2, 4 ) ( 2, 2, 10 ) ( 2, 6, 5 ) ( 5, 2, 5 ) ( 2, 3, 6 ) ( 2, 7, 1 ) ( 5, 3, 1 ) ( 2, 3, 7 ) ( 2, 7, 2 ) ( 5, 3, 2) ( 2, 3, 8 ) ( 2, 7, 3 ) ( 5, 3, 3 ) ( 2, 3, 9 ) ( 2, 3, 4 ) ( 2, 3, 5 ) (2 , 4, 6 ) ( 2 , 8, 1) ( 5, 4, 1 ) ( 2, 4, 7 ) ( 2, 8, 2 ) ( 5, 4, 2 ) ( 2, 4, 8 ) ( 2, 4, 3 ) ( 2, 4, 4 ) ( 2, 4, 5 ) ( 3, 1, 6 ) ( 3, 5, 1 ) ( 9, 1, 1 ) ( 3, 1, 7 ) ( 3, 5, 2 ) ( 6, 1, 2 ) ( 3, 1, 8 ) ( 3, 5, 3 ) ( 6, 1, 3 ) (3, 1, 9 ) ( 3, 5, 4 ) ( 6, 1, 4 )

Realization [ 4, 9, 1, 6, 11, 2, 7, 10, 5, 0, 8, 3 ] [ 5, 10, 3, 7, 0, 6, 2, 9, 1, 4, 8 ] [ 4, 7, 3, 0, 5, 8, 1, 6, 2, 9 ] [ 7, 12, 2, 10, 1, 9, 4, 0, 5, 8, 3, 11, 6 ] [ 6, 11, 3, 7, 2, 10, 1, 8, 4, 0, 9, 5 ] [ 3, 8, 0, 5, 2, 9, 1, 6, 10, 7, 4 ] [ 7, 12, 2, 11, 1, 10, 5, 0, 3, 8, 13, 4, 9, 6 ] [ 1, 9, 0, 5, 8, 4, 12, 3, 11, 2, 6, 10, 7 ] [ 4, 7, 10, 2, 5, 0, 9, 6, 1, 8, 3, 11 ] [ 10, 0, 5, 8, 13, 3, 7, 12, 2, 6, 1, 11, 14, 9, 4 ] [ 5, 9, 0, 4, 1, 10, 6, 2, 11, 8, 13, 3, 7, 12 ] [ 4, 0, 10, 2, 7, 12, 9, 1, 5, 8, 11, 6, 3 ] [ 4, 7, 11, 6, 1, 9, 2, 5, 0, 8, 3, 10 ] [ 4, 1, 8, 5, 0, 7, 3, 10, 6, 2, 9 ] [ 6, 9, 2, 5, 1, 8, 3, 7, 4, 0 ] [ 4, 9, 1, 6, 2, 10, 0, 5, 8, 12, 7, 3, 11 ] [ 5, 10, 2, 6, 9, 1, 8, 4, 0, 3, 7, 11 ] [ 5, 9, 1, 7, 2, 10, 6, 3, 0, 8, 4 ] [ 9, 0, 5, 2, 11, 6, 1, 10, 7, 3, 12, 8, 4, 13 ] [ 5, 10, 1, 11, 2, 6, 9, 0, 4, 8, 3, 12, 7 ] [ 4, 8, 1, 6, 9, 0, 5, 2, 11, 3, 7, 10 ] [ 6, 11, 1, 12, 2, 7, 10, 14, 9, 4, 0, 5, 8, 3, 13 ] [ 1, 6, 2, 7, 3, 0, 5, 8, 4, 9 ] [ 5, 10, 7, 2, 9, 1, 6, 0, 4, 8, 3 ] [ 10, 2, 6, 1, 11, 3, 7, 12, 9, 5, 0, 8, 4 ] [ 7, 11, 3, 6, 10, 2, 9, 1, 5, 8, 4, 0 ] [ 2, 5, 9, 1, 8, 0, 3, 7, 4, 10, 6 ] [ 6, 11, 2, 7, 12, 1, 10, 0, 4, 9, 13, 3, 8, 5 ] [ 10, 1, 6, 9, 0, 5, 2, 11, 7, 3, 12, 8, 4 ] [ 9, 6, 3, 0, 5, 1, 10, 2, 7, 11, 8, 4 ] [ 9, 14, 4, 7, 11, 6, 1, 5, 10, 0, 3, 13, 2, 12, 8 ] [ 7, 1, 5, 0, 4, 9, 2, 6, 3, 8 ] [ 3, 8, 2, 6, 10, 7, 0, 5, 1, 9, 4 ] [ 3, 7, 11, 8, 1, 6, 2, 10, 5, 0, 9, 4 ] [ 5, 10, 2, 7, 1, 6, 9, 4, 0, 8, 3 ] [ 9, 2, 6, 0, 4, 7, 3, 8, 5, 1 ] [ 11, 2, 5, 8, 0, 3, 6, 9, 4, 1, 10, 7 ] [ 10, 3, 8, 11, 6, 1, 9, 2, 5, 0, 7, 4 ] [ 4, 7, 0, 6, 2, 10, 3, 8, 1, 9, 5 ] [ 5, 2, 7, 0, 4, 1, 8, 3, 6, 9 ] [ 4, 7, 12, 9, 1, 6, 10, 2, 5, 0, 8, 3, 11 ] [ 4, 9, 5, 1, 10, 6, 11, 8, 0, 3, 7, 2 ] [ 3, 8, 5, 2, 10, 7, 4, 0, 6, 1, 9 ] [ 4, 9, 0, 3, 8, 13, 10, 5, 1, 12, 7, 2, 11, 6 ] [ 4, 8, 0, 5, 1, 11, 6, 2, 10, 7, 3, 12, 9 ] [ 11, 2, 7, 4, 1, 8, 3, 0, 9, 6, 10, 5 ]

.m

( 2, 3, 4 ) ( 3, 5, –) ( 4, 5,– ) ( 5, 6, 7 ) ( 4, 5, – ) ( 2, 3, – ) ( 5, 6, 7 ) (6, 7,– ) ( 8, 3,– ) (9, 3, 4 ) ( 7, 3, –) ( 2, 3,– ) ( 8, 3, 4 ) ( 7, 3,– ) (2, 3,– ) ( 9, 3, 4 ) ( 4, 5,– ) (4, 5,– ) ( 8, 9, 4 ) ( 4, 5, – ) ( 8, 3, – ) ( 11, 5, 6 ) (4, 5,–) ( 2, 3, 5 ) ( 9, 3, 4 ) ( 5, 7,–) ( 4, 6,–) ( 4, 5, 6 ) ( 8, 3, – ) ( 8, 3, – ) ( 6, 8, 9 ) (2, 3,– ) ( 2, 3, 4 ) (2, 3, 4 ) (2, 3, 5 ) (2, 3,–) ( 2, 7, – ) ( 8, 3, 4 ) ( 4, 5,–) ( 2, 3,– ) ( 8, 3, 4 ) ( 2, 4, –) ( 2, 3,– ) ( 2, 4, 6 ) ( 8, 3,–) (2, 5,–)

Method 3

3

3

3

3

3

3

2 1 3

3

2 1 1 3

3

3

3

422

P. Chand and M. A. Ollis

Table 8 Growable realizations for .{3a , 4b , 5c } separated by congruence modulo .(3, 4, 5): case to .(3, 4, 5). They are x-growable at .mx in .m = (m3 , m4 , m5 )

.(3, 1, 5)

Case (3,1,5)

(3,2,1)

(3,2,2)

(3,2,3)

(3,2,4)

(3,2,5) (3,3,1)

(3,3,2)

(3,3,3) (3,3,4) (3,3,5) (3,4,1)

(3,4,2) (3,4,3) (3,4,4) (3,4,5)

(a,b,c) ( 3, 1, 10) ( 3, 5, 5 ) ( 6, 1, 5 ) ( 3, 2, 6) ( 3, 6, 1) ( 6, 2, 1) ( 3, 2, 7 ) ( 3, 6, 2 ) ( 6, 3, 2 ) ( 3, 2, 8 ) ( 3, 6, 3 ) ( 6, 2, 3) ( 3, 2, 9) ( 3, 6, 4 ) ( 6, 2, 4 ) ( 3, 2, 5) ( 3, 3, 6 ) ( 3, 7, 1 ) ( 6, 3, 1 ) (3, 3, 7) ( 3, 7, 2 ) ( 6, 3, 2 ) ( 3, 3, 8 ) ( 3, 3, 3 ) ( 3, 3, 4 ) ( 3, 3, 5 ) ( 3, 4, 6) ( 3, 8, 1) ( 6, 4, 1) ( 3, 4, 7 ) ( 3, 4, 2 ) ( 3, 4, 3 ) ( 3, 4, 4 ) ( 3, 4, 5)

Realization [ 13, 3, 8, 11, 1, 6, 10, 0, 5, 2, 12, 7, 4, 14, 9 ] [ 10, 5, 8, 13, 9, 0, 11, 1, 6, 2, 12, 3, 7, 4 ] [ 5, 10, 0, 8, 3, 12, 2, 7, 4, 1, 11, 6, 9 ] [ 4, 7, 11, 6, 1, 8, 3, 10, 2, 5, 0, 9 ] [ 6, 2, 9, 1, 8, 4, 0, 3, 7, 10, 5 ] [ 2, 5, 1, 8, 4, 9, 6, 3, 0, 7 ] [ 6, 11, 3, 8, 0, 4, 7, 2, 12, 9, 1, 10, 5 ] [ 5, 1, 4, 8, 11, 3, 7, 0, 9, 2, 6, 10 ] [ 7, 0, 3, 6, 11, 8, 4, 1, 10, 2, 5, 9 ] [ 6, 11, 7, 2, 13, 4, 9, 0, 3, 8, 12, 1, 10, 5 ] [ 11, 1, 6, 10, 2, 5, 9, 0, 3, 7, 12, 8, 4 ] [ 11, 8, 5, 1, 10, 7, 0, 3, 6, 2, 9, 4 ] [ 3, 8, 13, 10, 5, 0, 12, 7, 11, 1, 6, 2, 14, 9, 4 ] [ 6, 11, 7, 2, 13, 9, 12, 1, 5, 10, 0, 4, 8, 3 ] [ 0, 3, 7, 10, 1, 9, 4, 12, 2, 5, 8, 11, 6 ] [ 3, 6, 1, 8, 0, 5, 10, 7, 2, 9, 4 ] [ 3, 8, 11, 7, 12, 2, 6, 1, 9, 4, 0, 10, 5 ] [ 3, 7, 11, 6, 2, 10, 1, 4, 0, 8, 5, 9 ] [ 9, 5, 8, 0, 4, 1, 7, 10, 2, 6, 3 ] [ 11, 2, 6, 1, 10, 13, 4, 9, 0, 3, 7, 12, 8, 5 ] [ 6, 10, 1, 9, 5, 2, 11, 8, 4, 0, 3, 12, 7 ] [ 1, 10, 7, 11, 2, 6, 9, 4, 0, 3, 8, 5 ] [ 9, 12, 2, 7, 10, 14, 4, 8, 13, 3, 6, 11, 1, 5, 0 ] [ 7, 3, 8, 2, 5, 0, 4, 1, 6, 9 ] [ 5, 10, 7, 2, 9, 4, 0, 8, 1, 6, 3 ] [ 6, 11, 2, 7, 0, 9, 5, 1, 10, 3, 8, 4 ] [ 2, 5, 0, 11, 6, 1, 10, 7, 3, 12, 8, 4, 13, 9 ] [ 7, 11, 3, 12, 2, 6, 9, 5, 1, 10, 0, 4, 8 ] [ 4, 7, 3, 0, 8, 11, 2, 6, 9, 1, 10, 5 ] [ 3, 8, 13, 10, 7, 12, 1, 5, 0, 11, 6, 2, 14, 9, 4 ] [ 5, 8, 4, 9, 2, 6, 1, 7, 3, 0 ] [ 4, 8, 0, 7, 3, 9, 1, 6, 2, 10, 5 ] [ 11, 7, 3, 10, 2, 5, 0, 9, 6, 1, 8, 4] [ 4, 7, 11, 3, 8, 12, 2, 5, 0, 9, 1, 6, 10 ]

.m

( 8, 9, 4 ) ( 3, 4,–) ( 3, 5,–) ( 8, 3, 4 ) ( 4, 5,–) (2, 4,–) (4, 5, 6) (8, 4,–) (2, 7,–) (4, 5, 6 ) ( 2, 4,–) (2, 4,–) ( 2, 3, 4 ) ( 2, 3,–) ( 4, 6,– ) ( 2, 3, 4 ) ( 2, 3, 5 ) ( 2, 3,–) ( 2, 3,–) ( 9, 4, 5 ) ( 6, 7,–) (4, 5,–) ( 8, 9, 4 ) ( 2, 3,–) ( 2, 3, 5 ) ( 3, 4, 6 ) ( 8, 9, 4 ) ( 6, 7,– ) ( 4, 5,–) (2, 3, 4) (4, 5,–) (3, 4, 5) ( 8, 3, 4) ( 8, 3, 4)

Method 3

3

3

3

3

1 3

3

2 1 1 3

2 1 1 1

1. Find an .{x, y}-growable realization for .(aI , bJ , c0 ) for some .I, J ≥ 0. 2. For each i with .0 ≤ i < I , find a y-growable realization for .(ai , bji , c0 ), unless .(ai , b, c0 ) is inadmissible for all b, and a realization for .(ai , bj , c0 ) for all admissible triples with .0 ≤ j < ji .

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3. For each j with .0 ≤ j < J , find an x-growable realization for .(aij , bj , c0 ), unless .(a, bj , c0 ) is inadmissible for all a, and a realization for .(ai , bj , c0 ) for all admissible triples with .I ≤ i < ij . If we complete these steps, we claim that we have now also covered all .(a, b, c) with .c = c0 : Consider admissible .(a, b, c) and set .c = c0 . If a and b are large (that is, .a ≥ aI and .b ≥ bJ ) then Item 1 implies the existence of the required realization. If a is small (that is, .a < aI ), then Item 2 implies the existence of the required realization. Finally, if b is small (that is, .b < bJ ) then we get the required realization from Item 3. The algorithm works as follows. Start by setting .(a ' , b' , c' ) with .0 < a ' ≤ x, ' ' .0 < b ≤ y and .0 < c ≤ z (that is, set these to the original .(a0 , b0 , c0 ) of ' ' ' Sect. 3). If .(a , b , c ) has a U -growable realization we have completed this case. If not, complete the above steps, and we have completed all cases with .c = c' . We may now begin again by looking for a U -growable realization for .(a ' , b' , c' + z) and continuing with the steps above if unsuccessful. There is no guarantee that the algorithm terminates. However, if it does, we have a finite list of realizations that proves that case of the conjecture for multisets with underlying set U . There are places in the algorithm where there are choices to be made. These choices can help the algorithm run more quickly (or terminate at all). The most important of these is that at each run through the steps we may choose which vertices of the underlying sets are playing the roles of x, y and z. We may always choose z to be (one of) the vertices that occurs least frequently in the multiset to avoid the possibility of only looking at the family .(a ' , b' , c+kz), which might never have a U growable realization. In fact, our implementation begins at the end by first finding a U -growable realization for some .(a, b, c) with v as small as possible. It then makes choices of orderings of x, y and z that head towards this known finishing point. Before turning to the algorithm to provide results, we give a theoretical construction that will make the process smoother. Lemma 15 Let .L = {1a , 2b , x}. If L is admissible then it is realizable. Proof By Corollary 10, Theorem 14 and the definition of edge-length we may assume that .6 < x ≤ v/2. We may also assume that .a, b ≥ 1, as the BHR Conjecture is settled for all two-element sets [3, 4], and that x is odd, as the BHR Conjecture is settled for .{1a , 2b , x c } when x is even and .a + b ≥ x − 1 [8]. We first define a sequence that has the multiset .{1, 2k−1 } as differences. If k is odd, let σk = 0, 2, 4, . . . , k − 1, k, k − 2, k − 4, . . . , 1

.

and if k is even let σk = 0, 2, 4, . . . , k, k − 1, k − 3, k − 5, . . . , 1.

.

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Denote the sequence obtained from .σk by adding y to each vertex as .σk + y. Given a sequence that ends with .k+1 consecutive numbers .y, y+1, . . . , y+k, we may replace those numbers with .σk + y to replace the .k − 1 of realized differences 1 with .k − 1 realized differences of 2. Consider the sequence v − 1, v − 2, . . . , x; 0, 1, . . . , x − 1.

.

This is a Hamiltonian path that realizes .{1v−2 , x} and either side of the semi-colon is a sequence of consecutive numbers. We may replace subsequences at one or both ends with sequences of the form .σk + y to obtain a sequence that realizes .{1a , 2b , x} for any a in the range .2 ≤ a ≤ v − 2. This leaves the case .a = 1. If v is even then the Hamiltonian path v−2, v−4, . . . , x+1, x−1, x−2, x−4, . . . , 1, v−1, v−3, . . . , x, 0, 2, 4, . . . , x−3

.

where all differences are 2 except between .x − 1 and .x − 2 and between x and 0, realizes .{1, 2v−3 , x}. If v is odd, the same is true of v−2, v−4, . . . , x+2, x, 0, 2, 4, . . . , x−1, x−2, x−4, . . . , 1, v−1, v−3, . . . , x+1.

.

⨆ ⨅ Example 16 Let .v = 16 and .x = 7. The following are some realizations of multisets of the form .{1a , 2b , 7} obtainable from the proof of Lemma 15. The initial Hamiltonian path that realizes .{114 , 7} is [15, 14, 13, 12, 11, 10, 9, 8, 7, 0, 1, 2, 3, 4, 5, 6].

.

To realize .{110 , 24 , 7}, we can replace the last six vertices with .σ5 + 1 as follows: [15, 14, 13, 12, 11, 10, 9, 8, 7, 0, 1, 3, 5, 6, 4, 2].

.

To realize .{17 , 27 , 7} we can replace the first five vertices of this new sequence with .σ4 + 11, reversed: [12, 14, 15, 13, 11, 10, 9, 8, 7, 0, 1, 3, 5, 6, 4, 2].

.

The Hamiltonian path that realizes .{1, 212 , 7} is [14, 12, 10, 8, 6, 5, 3, 1, 15, 13, 11, 9, 7, 0, 2, 4].

.

We can now use the algorithm to prove the main result of the section. Theorem 17 .BHR(L) holds for multisets L whose underlying set is one of the following 25 sets of size 3:

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{1, 2, 7}, {1, 2, 9}, {1, 3, 6}, {1, 3, 7}, {1, 3, 8}, {1, 4, 6}, {1, 4, 7}, {1, 5, 6},

.

{1, 5, 7}, {1, 6, 7}, {2, 3, 6}, {2, 3, 7}, {2, 4, 7}, {2, 5, 6}, {2, 5, 7}, {2, 6, 7},

.

{3, 4, 6}, {3, 4, 7}, {3, 5, 6}, {3, 5, 7}, {3, 6, 7}, {4, 5, 6}, {4, 5, 7}, {4, 6, 7}, {5, 6, 7}.

.

Proof We use the above algorithm, implemented in GAP and available on the ArXiv page for this paper or by request from the authors, to generate most of the required realizations to prove the result. The algorithm is unsuccessful as it stands for underlying sets of the form .{x, 3, 6}. In particular, .{3, 6}-growable realizations for multisets of the form .{3a , 6b , x} appear to not exist. We circumvent this issue by using Lemma 15 as follows. A multiset .L = {3a , 6b , x} is admissible only if .v = a + b + 1 + 1 is not a multiple of 3. In this case, division by .3 (mod v) is an automorphism of .Zv and applying it to L we get an equivalent problem for the multiset .L' = {1a , 2b , ±x/3}. This equivalent problem has a realization by Lemma 15. Hence when applying the algorithm to underlying sets of the form .{3, 6, x} we may assume that x appears at least twice in any multiset we need to investigate. This is sufficient for the algorithm to go through for the five cases of this form here. In the order of the statement of the theorem, the number of realizations produced by the program for each underlying set is 198, 578, 140, 263, 402, 226, 361, 296,

.

405, 542, 172, 310, 453, 314, 718, 576,

.

285, 567, 367, 598, 676, 390, 685, 808, 774

.

for a total of 11,104 realizations. The file containing all of the realizations required for this proof in GAP-readable format is also available on the ArXiv page or by request from the authors. ⨆ ⨅

6 Concluding Remarks We now have everything we need to prove the main result. Proof of Theorem 3 Let .U = {x, y, z} and .L = {x a , y b , zc }. If .max (U ) ≤ 5 then Corollary 10 gives the result. Consider U with .max (U ) = 6. The set .{1, 2, 6} is covered in [8]. If .U = {2, 4, 6} and v is even, then L is inadmissible. If v is odd then we may take the solution for a b c .{1 , 2 , 3 } from [2] and apply the .Zv -automorphism .t |→ 2t. Otherwise, U is covered by Theorem 17.

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All cases with .max (U ) = 7 are directly covered by Theorem 17. When max (U ) > 7 we have five cases to consider: .{1, 2, 8}, .{1, 2, 9}, .{1, 2, 10}, .{1, 2, 12} and .{1, 3, 8}. The set .{1, 2, 8} is covered in [8], the sets .{1, 2, 9} and .{1, 3, 8} are covered by Theorem 17, and the sets .{1, 2, 10} and .{1, 2, 12} are covered by Theorem 14. ⨆ ⨅

.

We have presented three approaches to completely solving instances of the BHR Conjecture for a given three-element underlying set. Each has its advantages and its drawbacks when considering how they might contribute to further progress on the problem. The approach of Sect. 3 has the advantage of being human-readable and allows insight into the structure of the problem at hand to reduce the number of realizations required for a particular underlying set. However, the number of realizations that are required is large and will quickly grow larger as the elements in the underlying sets do. More insight into how to construct growable realizations is needed to get anywhere near being able to implement the approach without resorting to a computer search. We believe that the approach of Sect. 4 offers the most hope for the most substantial theoretical advances: Coupling non-growable techniques with growable ones gives a way to harness other theoretical approaches and complement them with the theory of growable realizations. Of course, this requires the concomitant development of other theoretical approaches. The approach of Sect. 5 also offers hope. Might one prove that the steps of the algorithm are guaranteed to succeed, hence proving the conjecture for some underlying sets (perhaps only for a portion of the .π(U ) cases, or with a finite number of missing subcases) without needing to run the programs? If there is a specific three-element underlying set of interest, with elements that are not too large, then the Sect. 5 approach is probably the one that will most quickly lead to a resolution, given current tools. Acknowledgments We are grateful to Onur Agirseven, Kat Cannon-MacMartin, Eamon Mahoney, Anita Pasotti, Marco Pellegrini and John Schmitt for conversations about the BHR Conjecture. The majority of this work was completed in the summer of 2021 under the auspices of the School of Communication Research Co-Curricular Program at Emerson College. We are also grateful to the organizers of the 53rd Southeastern International Conference on Combinatorics, Graph Theory and Computing, especially for implementing a successful hybrid format without which this work could not have been presented. Finally, we are grateful for the suggestions of an anonymous referee, whose comments improved the presentation of the material.

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