Generalized Connectivity of Graphs [1st ed.] 3319338277, 978-3-319-33827-9, 978-3-319-33828-6, 3319338285

Table of contents :
Front Matter....Pages i-x
Introduction....Pages 1-13
Results for Some Graph Classes....Pages 15-29
Algorithm and Complexity....Pages 31-39
Sharp Bounds of the Generalized (Edge-)Connectivity....Pages 41-57
Graphs with Given Generalized Connectivity....Pages 59-66
Nordhaus-Gaddum-Type Results....Pages 67-77
Results for Graph Products....Pages 79-88
Maximum Generalized Local Connectivity....Pages 89-112
Generalized Connectivity for Random Graphs....Pages 113-133
Back Matter....Pages 135-143

Citation preview

SPRINGER BRIEFS IN MATHEMATICS

Xueliang Li Yaping Mao

Generalized Connectivity of Graphs

123

SpringerBriefs in Mathematics

Series Editors Nicola Bellomo Michele Benzi Palle E.T. Jorgensen Tatsien Li Roderick Melnik Otmar Scherzer Benjamin Steinberg Lothar Reichel Yuri Tschinkel G. George Yin Ping Zhang

SpringerBriefs in Mathematics showcases expositions in all areas of mathematics and applied mathematics. Manuscripts presenting new results or a single new result in a classical field, new field, or an emerging topic, applications, or bridges between new results and already published works, are encouraged. The series is intended for mathematicians and applied mathematicians. More information about this series at http://www.springer.com/series/10030

Xueliang Li • Yaping Mao

Generalized Connectivity of Graphs

123

Xueliang Li Center for Combinatorics Nankai University Tianjin, China

Yaping Mao Center for Combinatorics Nankai University Tianjin, China

ISSN 2191-8198 ISSN 2191-8201 (electronic) SpringerBriefs in Mathematics ISBN 978-3-319-33827-9 ISBN 978-3-319-33828-6 (eBook) DOI 10.1007/978-3-319-33828-6 Library of Congress Control Number: 2016939145 Mathematics Subject Classification (2010): 05C05, 05C35, 05C40, 05C70, 05C75, 05C76, 05C80, 05C85, 68M10, 68Q25, 68R10 © The Author(s) 2016 This work is subject to copyright. All rights are reserved 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, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. Printed on acid-free paper This Springer imprint is published by Springer Nature The registered company is Springer International Publishing AG Switzerland

Preface

Connectivity is one of the most basic concepts of graph-theoretical subjects, both in a combinatorial sense and in an algorithmic sense. As we know, the classical connectivity has two equivalent definitions, one is the “cut” version; the other is the “path” version. The generalized connectivity of a graph G is a natural generalization of the “path” version definition. For a graph G D .V; E/ and a set S  V.G/ of at least two vertices, an S-Steiner tree or a Steiner tree connecting S (or simply, an S-tree) is such a subgraph T D .V 0 ; E0 / of G that is a tree with S  V 0 . Note that when jSj D 2, a minimal Steiner tree connecting S is just a path connecting the two vertices of S. Two Steiner trees T and T 0 connecting S are said to be internally disjoint if E.T/ \ E.T 0 / D ¿ and V.T/ \ V.T 0 / D S. For S  V.G/ and jSj  2, the generalized local connectivity G .S/ is the maximum number of internally disjoint Steiner trees connecting S in G, that is, we search for the largest number of edgedisjoint Steiner trees which contain S and are vertex disjoint with the exception of the vertices in S. For an integer k with 2  k  n, the generalized k-connectivity (or k-tree-connectivity) is defined as k .G/ D minfG .S/ j S  V.G/; jSj D kg, that is, k .G/ is the minimum value of G .S/ when S runs over all k-subsets of V.G/. Clearly, when jSj D 2, 2 .G/ is just the connectivity .G/ of G, that is, 2 .G/ D .G/, which is the reason why one addresses k .G/ as the generalized connectivity of G. By convention, for a connected graph G with less than k vertices, we set k .G/ D 1, and k .G/ D 0 when G is disconnected. This concept was first mentioned in a paper by Hager published in 1985 and studied by himself in his other unpublished paper. So, it should have appeared before 1985. There are many other kinds of generalizations of the classical connectivity, such as pendant tree-connectivity, path-connectivity, k-connectivity, restricted connectivity, super connectivity, etc. But, they are different from the generalized k-connectivity we deal with in this book. As a natural counterpart of the generalized connectivity, we introduced the concept of generalized edge-connectivity k .G/, which is defined similarly to k .G/ only with the change of internally disjoint Steiner trees into edge-disjoint Steiner trees, i.e., only with E.T/ \ E.T 0 / D ¿ but without V.T/ \ V.T 0 / D S. The generalized edge-connectivity is related to two important problems. For a given v

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Preface

graph G and S  V.G/, the problem of finding a maximum set of edge-disjoint Steiner trees connecting S in G is called the Steiner tree packing problem. The problem for S D V.G/ is called the spanning-tree packing problem. Note that spanning tree packing problem is a specialization of the Steiner tree packing problem (for k D n, each Steiner tree connecting S is a spanning tree of G). For any graph G of order n, the spanning tree packing number or STP number is the maximum number of edge-disjoint spanning trees contained in G. From the definitions of k .G/ and k .G/, n .G/ D n .G/ is exactly the spanning-tree packing number of G (for k D n, internally disjoint Steiner trees connecting S are edgedisjoint spanning trees). Both extremal cases of k, i.e., k D 2 and n, for the two parameters are fundamental theorems in graph theory. The extremal case k D 2 of the problem means when we have two terminals. In this case internally (edge-)disjoint trees are just internally (edge-)disjoint paths between the two terminals, and so the problem becomes the well-known Menger theorem. The other extremal case k D n means when all the vertices are terminals. In this case, internally disjoint Steiner trees are just edge-disjoint spanning trees of the graph, and so the problem becomes the classical Nash-Williams-Tutte theorem. The generalized edge-connectivity and the Steiner tree packing problem have applications in VLSI circuit design. In this application, a Steiner tree is needed to share an electronic signal by a set of terminal nodes. A Steiner tree is also used in computer communication networks and optical wireless communication networks. Another application, which is our primary focus, arises in the Internet domain. Imagine that a given graph G represents a network. We choose arbitrary k vertices as nodes. Suppose one of the nodes in G is a broadcaster, and all other nodes are either users or routers (also called switches). The broadcaster wants to broadcast as many streams of movies as possible, so that the users have the maximum number of choices. Each stream of movie is broadcasted via a tree connecting all the users and the broadcaster. So, in essence, we need to find the maximum number of Steiner trees connecting all the users and the broadcaster, namely, we want to get .S/, where S is the set of the k nodes. Clearly, it is a Steiner tree packing problem. Furthermore, if we want to know whether for any k nodes the network G has the above properties, then we need to compute k .G/ D minf.S/g in order to prescribe the reliability and the security of the network. After the concept of generalized connectivity was proposed by Hager in 1985, the study on this graph parameter kept silent for about 25 years. Only in recent 5 years when a paper by Chartrand et al. in 2010 reintroduced it, this parameter then caught people’s attention. As a result, the study on this parameter was stimulated, and there have been quite a lot of results published. The goal of this book is to bring together most of the results that deal with it. We begin with an introductory chapter. In Chap. 2, we summarize the results on the generalized (edge-)connectivity for some graph classes, giving the reader some intuitive idea about the parameter. In Chap. 3, we address the problem of algorithms and computational complexity for the generalized (edge-)connectivity. In general, it is NP-hard. Chapter 4 is then to report sharp bounds that have been obtained in recent years. Chapter 5 is to characterize

Preface

vii

the graphs with large generalized (edge-)connectivity, and Chap. 6 is to develop Nordhaus-Gaddum-type inequalities. Some results on graph operations are reported in Chap. 7. Extremal problems on the generalized local (edge-) connectivity are reported in Chap. 8. The last chapter, Chap. 9, covers results for random graphs. In each chapter, we list some conjectures, open problems, or questions at proper places. We hope that this can motivate more young graph theorists and graduate students to do further study in this subject. We do not give proofs for all results. Instead, we only select some of them for which we gave their proofs because we feel that these proofs employed some typical techniques, and these proof techniques are popular in the study of generalized (edge-)connectivity. New results are still appearing. There must be some or even many of them for which we have not realized their existence and therefore have not included them in this book. The readers of the book are expected to have some background in graph theory and some related knowledge in combinatorics, probability, algorithms, and complexity analysis. All relevant notions from graph theory are properly defined in Chap. 1, but also elsewhere where needed. The anticipated readers of the book are mathematicians and students of mathematics, whose fields of interest are graph theory and combinatorial optimization. Consequently, this book will be found suitable for such courses. The exposition of the details of the proofs of some main results will enable students to understand and eventually master a good part of graph theory and combinatorial optimization. People working on communication networks may also be interested in some aspects of the book. They will find it useful for designing networks that can efficiently, reliably, and safely transfer information. The material presented in this book was used in graph theory seminars, held three times at Nankai University, in 2013, 2014, and 2015. We thank all the members of our group for help in the preparation of this book. Without their help, we would have not finished writing it in such a short period of time. We also thank the NSFC (National Natural Science Foundation of China) for financial support to our research project on generalized (edge-)connectivity. Last but not least, we are very grateful to the editor for algebraic combinatorics and graph theory of this new series of books of Springer Briefs, Professor Ping Zhang, for inviting us to write this book. Without her encouragement, this book may not exist. Tianjin, China

Xueliang Li Yaping Mao

Contents

1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Basic Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Connectivity and Its Generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.1 k-Connectivity and k-Edge-Connectivity . . . . . . . . . . . . . . . . . . . . . . 1.2.2 Generalized k-Connectivity and Generalized k-Edge-Connectivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.3 Mader’s Generalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.4 Pendant Tree-Connectivity and Path-Connectivity . . . . . . . . . . . . 1.3 Generalized Connectivity and Steiner Tree Packing Problem . . . . . . . . 1.4 Theoretical and Application Backgrounds of the Generalized Connectivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5 Generalized Edge-Connectivity, Edge Toughness, Strength, and Arboricity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.6 Generalized Connectivity and Rainbow Indices . . . . . . . . . . . . . . . . . . . . . . .

1 1 3 4

11 12

2

Results for Some Graph Classes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Results for the Spanning-Tree Packing Number. . . . . . . . . . . . . . . . . . . . . . . 2.2 Results for the Generalized k-Connectivity . . . . . . . . . . . . . . . . . . . . . . . . . . . .

15 15 20

3

Algorithm and Complexity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Basic Notions of NP-Completeness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Some Conjectures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

31 31 32 38

4

Sharp Bounds of the Generalized (Edge-)Connectivity . . . . . . . . . . . . . . . . . . 4.1 Bounds of the Generalized Connectivity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Bounds of the Generalized Edge-Connectivity . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Bounds for Planar and Line Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

41 41 51 55

5

Graphs with Given Generalized Connectivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Graphs with the Largest Generalized Connectivity . . . . . . . . . . . . . . . . . . . 5.2 Graphs with the Second Largest Generalized Connectivity . . . . . . . . . . .

59 59 63

5 7 7 9 9

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5.3 The Minimal Size of a Graph with Given Generalized k-Connectivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

64

6

Nordhaus-Gaddum-Type Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 Results for Graphs in G .n/ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Results for Graphs in G .n; m/ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

67 67 75

7

Results for Graph Products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1 Results for the Generalized 3-Connectivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Results for the Spanning-Tree Packing Number. . . . . . . . . . . . . . . . . . . . . . .

79 80 83

8

Maximum Generalized Local Connectivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 8.1 Results for the Maximum Generalized Local Connectivity . . . . . . . . . . . 90 8.2 Graphs with at Most Two Internally Disjoint Steiner Trees Connecting Any Three Vertices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 8.3 Graphs with at Most ` Internally Disjoint Steiner Trees Connecting Any n  1 Vertices. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

9

Generalized Connectivity for Random Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.1 Results for the Spanning-Tree Packing Number. . . . . . . . . . . . . . . . . . . . . . . 9.1.1 Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.1.2 Deterministic Tools . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.1.3 Properties of G.n; p/ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2 Results for the Generalized 3-Connectivity . . . . . . . . . . . . . . . . . . . . . . . . . . . .

113 114 115 116 117 126

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141

Chapter 1

Introduction

1.1 Basic Concepts In this section, we will bring together most of the terminology and notation used in this monograph. For those not given here, they will be defined whenever needed. All graphs considered in this book are finite, simple, and undirected, unless otherwise stated. We follow the graph theoretical terminology and notation of [19, 20] for all those not defined here. A graph G is denoted by an ordered pair .V.G/; E.G// where V.G/ is the set of vertices of G and E.G/ is the set of edges of G. If there is an edge uv between a vertex u and a vertex v, we say that u and v are adjacent and they are incident with the edge uv; we also say that v is a neighbor of u and u is a neighbor of v. The set of neighbors of a vertex v in a graph G is denoted by NG .v/. If there is more than one edge between u and v, we say that uv is a multiple edge. If a graph has multiple edges, it is called a multigraph. A uu edge is called a loop at the vertex u. If a graph does not have any loops or multiple edges, it is called a simple graph. A graph G0 is called a subgraph of a graph G if V.G0 /  V.G/ and E.G0 /  E.G/. For any subset X of V.G/, let GŒX denote the subgraph induced by X and EŒX the edge set of GŒX; similarly, for any subset F of E.G/, let GŒF denote the subgraph induced by F. We use G  X to denote the subgraph of G obtained by removing all the vertices of X together with the edges incident with them from G; similarly, we use G n F to denote the subgraph of G obtained by removing all the edges of F from G. If X D fvg and F D feg, we simply write G  v and G n e for G  fvg and G n feg, respectively. For two subsets X and Y of V.G/, we denote by EG ŒX; Y the set of edges of G with one end in X and the other end in Y. If X D fxg, we simply write EG Œx; Y for EG Œfxg; Y. A complete graph is a graph in which every pair of vertices is adjacent, and a complete graph on n vertices is denoted by Kn . A graph is bipartite if its vertex set can be partitioned into two subsets X and Y so that every edge has one end in X and the other end in Y; such a partition .X; Y/ is called a bipartition of the graph, and © The Author(s) 2016 X. Li, Y. Mao, Generalized Connectivity of Graphs, SpringerBriefs in Mathematics, DOI 10.1007/978-3-319-33828-6_1

1

2

1 Introduction

X and Y are its parts. If each vertex in X is adjacent to every vertex in Y, then G is called a complete bipartite graph. Let Ks;t denote a complete bipartite graph the cardinalities of whose two parts are s; t, respectively. A path on n vertices is denoted by Pn , and a cycle on n vertices is denoted by Cn . A connected graph without any cycles is called a tree. A forest is a graph whose every component is a tree. A subtree of a graph is a subgraph of the graph which is a tree. If this subtree is a spanning subgraph, it is called a spanning tree of the graph. The degree of a vertex v in a graph G, denoted by dG .v/, is the number of edges of G incident with v. A vertex of degree 0 is called an isolated vertex. We denote by ı.G/ and .G/ the minimum and maximum degrees of the vertices of G. A graph G is called k-regular if dG .v/ D k for every v 2 V.G/. A 3-regular graph is called a cubic graph. The distance between two vertices u and v in a connected graph G, denoted by dG .u; v/, is the length of a shortest path between them in G. The eccentricity of a vertex v is ecc.v/ WD maxx2V.G/ d.v; x/. The diameter of G is diam.G/ WD maxx2V.G/ ecc.x/. Two graphs are called disjoint if they have no vertex in common and edgedisjoint if they have no edge in common. The union G [ H of two graphs G and H is the graph with vertex set V.G/ [ V.H/ and edge set E.G/ [ E.H/. If G and H are disjoint, we refer to their union as a disjoint union. It can be seen that every graph may be expressed uniquely (up to order) as a disjoint union of connected graphs, and these graphs are called the connected components or simply components, and we denote the number of components of a graph G by !.G/. If G is the disjoint union of k copies of a graph H, we simply write G D kH. The joint G _ H of two disjoint graphs G and H is the graph with vertex set V.G/ [ V.H/ and edge set E.G/ [ E.H/ [ fuv j u 2 V.G/; v 2 V.H/g. The Cartesian product GH of two graphs G and H is the graph whose vertex set is V.G/V.H/ and whose edge set is the set of pairs .u; v/.u0 ; v 0 / such that either uu0 2 E.G/ and v D v 0 or vv 0 2 E.H/ and u D u0 . A k-dimensional hypercube, denoted by Qk , is the Cartesian product of k P2 s. The composition (lexicographic product) G ı H of two graphs G and H is the graph with vertex set V.G/  V.H/ in which .u; v/ is adjacent to .u0 ; v 0 / if and only if either uu0 2 E.G/ or u D u0 and vv 0 2 E.H/. Note that the Cartesian product is commutative, but the lexicographic product is noncommutative in general. The line graph L.G/ of a graph G is the graph whose vertex set is V.L.G// D E.G/ and two vertices e1 , e2 of L.G/ are adjacent if and only if they are adjacent in G. The iterated line graph of a graph G, denoted by L2 .G/, is the line graph of the graph L.G/. More generally, the k-iterated line graph Lk .G/ is the line graph of Lk1 .G/ .k  2/. The complement G of a graph G is the graph whose vertex set is V.G/ and whose edges are the pairs of nonadjacent vertices of G. A subdivision of G is a graph obtained from G by replacing some edges with pairwise internally disjoint paths. A graph G is planar if it can be embedded in the plane. We say that an edge of a graph G is a crossing edge if G is nonplanar and G n e is planar. Let G a planar graph embedded such that no edges cross each other in the plane. By a region of G, we mean a connected component of the complement of the set of points in the plane that belong to the vertices and edges of G.

1.2 Connectivity and Its Generalizations

3

1.2 Connectivity and Its Generalizations Connectivity is one of the most basic concepts of graph-theoretic subjects, both in a combinatorial sense and an algorithmic sense. As we know, the classical connectivity has two equivalent definitions; one is called the “path version,” and the other is called the “cut version.” The “path version” definition is as follows. Let G be a graph. For a pair of distinct vertices x and y of G, define a parameter G .x; y/ as the maximum number of internally vertex-disjoint paths between x and y in G. This parameter is addressed as the local connectivity of x and y in G, which is the “path version” definition of the local connectivity. Then, for the graph G, we get a global quantity .G/ D minfG .x; y/ j x; y 2 V.G/; x ¤ yg, which is the “path version” definition of the connectivity of G, whereas the “cut version” definition is as follows. For a pair of nonadjacent vertices x and y of G, a vertex subset S of V.G/  fx; yg is called a separator of x and y in G if x and y belong to different components of G  S. For a pair of distinct vertices x and y in G, define a parameter G .x; y/ to be the order of a minimum separator of x and y in G if xy … E.G/ and, otherwise, if xy 2 E.G/, G .x; y/ to be 1 plus the order of a minimum separator of x and y in G  xy. This parameter is called the “cut version” definition of the local connectivity. Then, for the graph G, we get a global quantity   .G/ D minfG .x; y/ j x; y 2 V.G/; x ¤ yg, which is the “cut version” definition of the connectivity of G. Actually,   .G/ is equal to the order of a minimum vertex subset S of V.G/ such that G  S is disconnected or has only one vertex. The usual form of Menger’s theorem [106] states that for any pair of distinct vertices x and y of a graph G, G .x; y/ D G .x; y/; see Theorem 2.2 in [14], for example. This implies that for a graph G, .G/ D   .G/. Whitney in [138] showed that for a noncomplete graph G, .G/ D minfG .x; y/ j x; y 2 V.G/; xy … E.G/g. In contrast to the parameter .G/ or   .G/, Bollobás introduced another parameter .G/ D maxfG .x; y/ j x; y 2 V.G/; x ¤ yg; see [14], for example, which is called the maximum local connectivity of G. As we have seen, the connectivity and maximum local connectivity are two extremes of the local connectivity of a graph. An invariant lying between these two extremes is theP average connectivity .G/ O  of a graph of order n, which is defined to be .G/ O D x;y2V.G/;x¤y G .x; y/= n2 ; see [10]. Similarly, the classical edge-connectivity also has two equivalent definitions; one is the “path version,” and the other is the “cut version.” The “path version” definition is as follows. For a graph G and a pair of distinct vertices x and y of G, define a parameter G .x; y/ as the maximum number of edge-disjoint paths between x and y in G. This parameter is addressed as the local edge-connectivity of x and y in G, which is the “path version” definition of the local edge-connectivity. Then, for the graph G, we get a global quantity .G/ D minfG .x; y/ j x; y 2 V.G/; x ¤ yg, which is the “path version” definition of the edge-connectivity of G, whereas the “cut version” definition is as follows. For a pair of distinct vertices x and y of G, an edge subset M of E.G/ is called an xy edge cut in G if x and y lie in different components of G n M. Then, define a parameter G .x; y/ as the order of a minimum xy edge cut in G. This parameter is called the “cut version” definition

4

1 Introduction

of the local edge-connectivity. Then, for the graph G, we get a global quantity  .G/ D minfG .x; y/ j x; y 2 V.G/; x ¤ yg, which is the “cut version” definition of the edge-connectivity of G. The edge version of Menger’s theorem states that for any pair of distinct vertices x and y of a graph G, G .x; y/ D G .x; y/; see [14, 20], for example. This implies that for a graph G, .G/ D  .G/. There are several nearly simultaneous discoveries about the edge version of Menger’s theorem, which appeared in papers by Ford and Fulkerson [43] (as a special case of their “max-flow, min-cut” theorem) and Elias, Feinstein, and Shannon [40] and also in unpublished work of Kotzig [59]. Also similarly, .G/ D maxfG .x; y/ j x; y 2 V.G/; x ¤ yg  P O and .G/ D x;y2V.G/;x¤y G .x; y/= n2 were introduced as the maximum local edgeconnectivity and average edge-connectivity, respectively. For more on the connectivity and edge-connectivity, we refer to Oellermann’s survey paper [110].

1.2.1 k-Connectivity and k-Edge-Connectivity Although there are many elegant and powerful results on connectivity in graph theory, the classical connectivity and edge-connectivity cannot be satisfied considerably in practical uses. So people tried to generalize these concepts for different purposes. For the “cut” version definition of connectivity, one can see that the above minimum vertex set does not consider the number of components of G  S. Two graphs with the same connectivity may have different degrees of vulnerability in the sense that the deletion of a vertex cutset of minimum cardinality from one graph may produce a graph with considerably more components than in the case of the other graph. For example, the star K1;n and the path PnC1 .n  3/ are both trees of order n C 1 and therefore have connectivity 1, but the deletion of a cut vertex from K1;n produces a graph with n components, while the deletion of a cut vertex from PnC1 produces only two components. The above statement suggests a generalization of the connectivity of a graph. In 1984, Chartrand et al. [27] generalized the “cut” version definition of connectivity. For an integer k .k  2/ and a graph G of order n .n  k/, the k-connectivity k0 .G/ is the smallest number of vertices whose removal from G produces a graph with at least k components or a graph with fewer than k vertices. Thus, for k D 2, 20 .G/ D .G/. For more details about the k-connectivity, we refer to [27, 37, 110, 111, 130]. If two graphs have the same edge-connectivity, then the removal of an edge set of minimum cardinality from either graph produces exactly two components. On the other hand, disconnecting these graphs into three components may require the removal of considerably more edges in one case than the other. Take, for example, if H1 is obtained from two copies of complete graph Kn .n  2/ by joining two vertices (one from each copy of Kn ) by an edge and H2 is a path of order 2n, then both graphs have order 2n and edge-connectivity 1. However, n edges need to be removed from H1 but only two edges from H2 to produce a graph with three components. This observation suggests a generalization of the “cut” version

1.2 Connectivity and Its Generalizations

5

definition of classical edge-connectivity. For an integer k .k  2/ and a graph G of order n .n  k/, the k-edge-connectivity 0k .G/ is the smallest number of edges whose removal from G produces a graph with at least k components. Thus, for k D 2, 02 .G/ D .G/. The k-edge-connectivity was initially introduced by Boesch and Chen [13] and subsequently studied by Goldsmith et al. in [48–50]. In all these papers, the computational difficulty of finding 0k .G/ for k  3 leads to the development of heuristics and bounds for approximating this parameter. For more details on the k-edge-connectivity, we refer to [9, 109].

1.2.2 Generalized k-Connectivity and Generalized k-Edge-Connectivity The generalized connectivity of a graph G, introduced by Hager [55], is a natural generalization of the “path” version definition of connectivity. For a graph G D .V; E/ and a set S  V.G/ of at least two vertices, an S-Steiner tree or a Steiner tree connecting S (or simply an S-tree) is a such subgraph T D .V 0 ; E0 / of G that is a tree with S  V 0 . Note that when jSj D 2, a minimal Steiner tree connecting S is just a path connecting the two vertices of S. Two Steiner trees T and T 0 connecting S are said to be internally disjoint if E.T/ \ E.T 0 / D ¿ and V.T/ \ V.T 0 / D S. For S  V.G/ and jSj  2, the generalized local connectivity G .S/ is the maximum number of internally disjoint Steiner trees connecting S in G, that is, we search for the largest number of edge-disjoint Steiner trees which contain S and are vertex disjoint with the exception of the vertices in S. For an integer k with 2  k  n, the generalized k-connectivity (or k-tree-connectivity) is defined as k .G/ D minfG .S/ j S  V.G/; jSj D kg, that is, k .G/ is the minimum value of G .S/ when S runs over all k-subsets of V.G/. Clearly, when jSj D 2, 2 .G/ is just the connectivity .G/ of G, that is, 2 .G/ D .G/, which is the reason why one addresses k .G/ as the generalized connectivity of G. By convention, for a connected graph G with less than k vertices, we set k .G/ D 1 and k .G/ D 0 when G is disconnected. Note that the generalized k-connectivity and the k-connectivity of a graph are indeed different. Take, for example, the graph G0 obtained from a triangle with vertex set fv1 ; v2 ; v3 g by adding three new vertices u1 ; u2 ; u3 and joining vi to ui by an edge for 1  i  3. Then 3 .G0 / D 1 but 30 .G0 / D 2. We knew the concept of generalized k-connectivity in [28] for the first time. There the authors obtained the exact value of the generalized k-connectivity of complete graphs. Later, from [55, 56], we learned that the concept was introduced actually by Hager in his another paper, yet we do not know whether this paper has been published. For results on the generalized connectivity (or tree-connectivity), we refer to [28, 30, 31, 47, 53, 74–86, 88–91, 96, 97, 112, 130, 131]. The following Table 1.1 shows how the generalization proceeds. As a natural counterpart of the generalized connectivity, we introduced the concept of generalized edge-connectivity in [90]. For S  V.G/ and jSj  2, the

6

1 Introduction Table 1.1 Classical connectivity and generalized connectivity Vertex subset

Set of Steiner trees

Local parameter Global parameter

Classical connectivity S D fx; yg  V.G/ .jSj D 2/ 8 Px;y D fP1 ; P2 ;    ; P` g ˆ ˆ ˆ ˆ ˆ < fx; yg  V.Pi /

Generalized connectivity S  V.G/ .jSj  2/ 8 TS D fT1 ; T2 ;    ; T` g ˆ ˆ ˆ ˆ ˆ < S  V.Ti /

ˆ E.Pi / \ E.Pj / D ¿ ˆ ˆ ˆ ˆ : V.P / \ V.P / D fx; yg i j

ˆ E.Ti / \ E.Tj / D ¿; ˆ ˆ ˆ ˆ : V.T / \ V.T / D S i j

.x; y/ D max jPx;y j .G/ D min .x; y/

.S/ D max jTS j k .G/ D min

x;y2V.G/

SV.G/;jSjDk

.S/

Table 1.2 Classical edge-connectivity and generalized edge-connectivity Vertex subset Set of Steiner trees

Edge-connectivity S D fx; yg  V.G/ .jSj D 2/ 8 Px;y D fP1 ; P2 ;    ; P` g ˆ ˆ < fx; yg  V.Pi / ˆ ˆ : E.P / \ E.P / D ¿

Generalized edge-connectivity S  V.G/ .jSj  2/ 8 TS D fT1 ; T2 ;    ; T` g ˆ ˆ < S  V.Ti /; ˆ ˆ : E.T / \ E.T / D ¿

.x; y/ D max jPx;y j .G/ D min .x; y/

.S/ D max jTS j k .G/ D min

i

Local parameter Global parameter

j

x;y2V.G/

i

j

SV.G/;jSjDk

.S/

generalized local edge-connectivity .S/ is the maximum number of edge-disjoint Steiner trees connecting S in G. For an integer k with 2  k  n, the generalized k-edge-connectivity k .G/ of G is then defined as k .G/ D minf.S/ j S  V.G/ and jSj D kg. It is also clear that when jSj D 2, 2 .G/ is just the standard edge-connectivity .G/ of G, that is, 2 .G/ D .G/, which is the reason why we address k .G/ as the generalized edge-connectivity of G. Also set k .G/ D 0 when G is disconnected. Results on the generalized edge-connectivity can be found in [86, 89–91, 96]. The above Table 1.2 shows how the generalization of the edge-version definition proceeds. Remark 1.2.1. The difference between the “path” version generalized connectivity k .G/ and the “cut” version k-connectivity k0 .G/ was discussed very clearly by Sun and Li in [130], where they got sharp lower and upper bounds for the difference k0 .G/  k .G/ and investigated the problem that under what conditions for a graph G, one has k0 .G/ D k .G/.

1.2 Connectivity and Its Generalizations

7

1.2.3 Mader’s Generalization In fact, Mader [103] studied an extension of Menger’s theorem [106] to independent sets of three or more vertices. We know that from Menger’s theorem that if S D fu; vg is a set of two independent vertices in a graph G, then the maximum number of internally disjoint u-v paths in G equals the minimum number of vertices that separate u and v. For a set S D fu1 ; u2 ;    ; uk g of k .k  2/ vertices in a graph G, an S-path is defined as a path between a pair of vertices of S that contains no other vertices of S. Two S-paths P1 and P2 are said to be internally disjoint if they are vertex disjoint except for the vertices of S. If S is a set of independent vertices of a graph G, then a vertex set U  V.G/ with U \ S D ¿ is said to totally separate S if every two vertices of S belongs to different components of G  U. Let S be a set of at least three independent vertices in a graph G. Let .G/ denote the maximum number of internally disjoint S-paths and 0 .G/ the minimum number of vertices that totally separate S. A natural extension of Menger’s theorem may well be suggested, namely, if S is a set of independent vertices of a graph G and jSj  3, then .S/ D 0 .S/. However, the statement is not true in general. Take the above graph G0 , for example. For S D fu1 ; u2 ; u3 g, .S/ D 1 but 0 .S/ D 2. Mader proved that .S/  12 0 .S/. Moreover, the bound is sharp. Lovász conjectured an edge analogue of this result, and Mader proved this conjecture and established its sharpness. For more details, we refer to [103, 104, 109].

1.2.4 Pendant Tree-Connectivity and Path-Connectivity Aside from the concept of tree-connectivity, Hager also introduced another treeconnectivity parameter, called the pendant tree-connectivity of a graph in [55]. For the tree-connectivity (or generalized connectivity), we only search for edge-disjoint trees which include S and are vertex disjoint with the exception of the vertices in S. But pendant tree-connectivity further requests the degree of each vertex of S in a Steiner tree connecting S is equal to one. Note that it is a specialization of the generalized connectivity (or tree-connectivity), but it is a generalization of the classical connectivity. The detailed definitions are stated as follows. For an S-Steiner tree, if the degree of each vertex in S is equal to one, then this tree is called a pendant S-Steiner tree. Two pendant S-Steiner trees T and T 0 are said to be internally disjoint if E.T/\E.T 0 / D ¿ and V.T/\V.T 0 / D S. For S  V.G/ and jSj  2, the pendanttree local connectivity G .S/ is the maximum number of internally disjoint pendant S-Steiner trees in G. For an integer k with 2  k  n, the pendant-tree k-connectivity is defined as k .G/ D minfG .S/ j S  V.G/; jSj D kg. Set k .G/ D 0 when G is disconnected. It is clear that  k .G/ D k .G/; for k D 1; 2I k .G/  k .G/; for k  3:

8

1 Introduction

Dirac [38] showed that in a .k  1/-connected graph, there is a path through any set of k vertices. Related problems were considered in [139]. In [56], Hager revised this statement to the question of how many internally disjoint paths Pi with the exception of a given set S of k vertices exist such that S  V.Pi /. Another concept of connectivity, the path-connectivity, of a graph G was also introduced by Hager in [56], which is also a specialization of the generalized connectivity and also a generalization of the “path” version definition of the classical connectivity. For a set S  V.G/ of at least two vertices, a Steiner path connecting S or an S-Steiner path (simply an S-path) is a subgraph P D .V 0 ; E0 / of G that is a path with S  V 0 . Note that an S-path is also a tree connecting S. Two S-paths P and P0 are said to be internally disjoint if E.P/ \ E.P0 / D ¿ and V.P/ \ V.P0 / D S. For S  V.G/ and jSj  2, the path local connectivity G .S/ is the maximum number of internally disjoint S-paths in G. For an integer k with 2  k  n, the path k-connectivity of a graph G on n vertices is defined as k .G/ D minfG .S/ j S  V.G/; jSj D kg. Clearly, we have 8 < k .G/ D ı.G/; for k D 1I  .G/ D .G/; for k D 2I : k k .G/  k .G/; for k  3: The relations between the pendant tree-connectivity, generalized connectivity, and path-connectivity are shown in the following Table 1.3. There are many other kinds of generalizations of the classical connectivity and edge-connectivity, such as the restricted (edge-)connectivity in [41] and super edge-connectivity in [99]. However, our intention of this book is to only focus on the generalized (edge-)connectivity. In very rare occasions, we will mention some results on the most closely related concepts: pendant tree-connectivity, pathconnectivity, and k-connectivity, in order to show the differences among them.

Table 1.3 Three kinds of tree-connectivities Pendant tree-connectivity S  V.G/ .jSj  2/ 8 TS D fT1 ; T2 ;    ; T` g ˆ ˆ ˆ < S  V.Ti / ˆ d Ti .v/ D 1 for every v 2 S ˆ ˆ : E.Ti / \ E.Tj / D ¿ .S/ D max jTS j k .G/ D min .S/ SV.G/;jSjDk

Generalized connectivity S  V.G/ .jSj  2/ 8 ˆ < TS D fT1 ; T2 ;    ; T` g S  V.Ti / ˆ : E.Ti / \ E.Tj / D ¿

Path-connectivity S  V.G/ .jSj  2/ 8 ˆ < PS D fP1 ; P2 ;    ; P` g S  V.Pi / ˆ : E.Pi / \ E.Pj / D ¿

.S/ D max jTS j k .G/ D min

.S/ D max jPS j k .G/ D min

SV.G/;jSjDk

.S/

SV.G/;jSjDk

.S/

1.4 Theoretical and Application Backgrounds of the Generalized Connectivity

9

1.3 Generalized Connectivity and Steiner Tree Packing Problem The generalized edge-connectivity is related to two important problems. For a given graph G and S  V.G/, the problem of finding a maximum set of edge-disjoint Steiner trees connecting S in G is called the Steiner tree packing problem. The difference between the Steiner tree packing problem and the generalized edgeconnectivity is as follows: the Steiner tree packing problem studies local properties of graphs since S is given beforehand, but the generalized edge-connectivity focuses on global properties of graphs since it first needs to compute the maximum number .S/ of edge-disjoint trees connecting S and then S runs over all k-subsets of V.G/ to get the minimum value of .S/. The problem for S D V.G/ is called the spanning-tree packing problem. Note that spanning-tree packing problem is a specialization of Steiner tree packing problem (for k D n, each Steiner tree connecting S is a spanning tree of G). For any graph G of order n, the spanning-tree packing number or STP number is the maximum number of edge-disjoint spanning trees contained in G. From the definitions of k .G/ and k .G/, n .G/ D n .G/ is exactly the spanning-tree packing number of G (for k D n, both internally disjoint Steiner trees connecting S and edge-disjoint Steiner trees connecting S are edge-disjoint spanning trees). For the spanning-tree packing number, we refer to [54, 113, 114]. Observe that spanningtree packing problem is a special case of both the generalized k-connectivity and the generalized k-edge-connectivity. This problem has two practical applications. One is to enhance the ability of fault tolerance [44, 61]. Consider a source node u that wants to broadcast a message on a network with ` edge-disjoint spanning trees. The node u copies ` messages to different spanning trees. If there are no more than `  1 fault edges, all the other nodes can receive the message. The other application is to develop efficient collective communication algorithms in distributed memory parallel computers [6, 98, 134]. If the above source node has a large number of data to transmit, we can let every edge-disjoint spanning tree be responsible for only 1=` data to increase the throughput. For any graph G, the maximum number of edge-disjoint spanning trees in G can be found in polynomial time; see [124], page 879. Actually, Roskind and Tarjan [122] proposed an O.m2 / time algorithm for finding the maximum number of edge-disjoint spanning trees in an arbitrary graph, where m is the number of edges in the graph.

1.4 Theoretical and Application Backgrounds of the Generalized Connectivity In addition to being natural combinatorial measures, the generalized connectivity and generalized edge-connectivity can be motivated by their interesting interpretation in practice as well as theoretical consideration.

10

1 Introduction

From a theoretical perspective, both extremes of this problem are fundamental theorems in combinatorics. One extreme of the problem is when we have two terminals. In this case internally (edge-)disjoint Steiner trees are just internally (edge-) disjoint paths between the two terminals, and so the problem becomes the well-known Menger’s theorem. The other extreme is when all the vertices are terminals. In this case internally disjoint Steiner trees and edge-disjoint Steiner trees are just edge-disjoint spanning trees of the graph, and so the problem becomes the classical Nash-Williams-Tutte theorem. Theorem 1.4.1 (Nash-Williams [107], Tutte [133]). A multigraph G contains a system of k edge-disjoint spanning trees if and only if kG=Pk  k.jPj  1/ holds for every partition P of V.G/, where kG=Pk denotes the number of edges in G between distinct blocks of P. The next theorem is due to Nash-Williams. Theorem 1.4.2 ([108]). Let G be a graph. Then the edge set of G can be covered by t forests if and only if, for every nonempty subset S of vertices of G, jEG ŒSj  t.jSj  1/. The following corollary can be easily derived from Theorem 1.4.1. Corollary 1.4.3. Every 2`-edge-connected graph contains a system of ` edgedisjoint spanning trees. Kriesell [64] conjectured that this corollary can be generalized for Steiner trees. Conjecture 1.4.4 (Kriesell [64]). A vertex subset S of a graph G is called `-edgeconnected, if the local edge-connectivity G .x; y/  ` for every pair of vertices x ¤ y in S. If a set S of vertices in a graph G is 2k-edge-connected, then there is a set of k edge-disjoint S-Steiner trees in G. Motivated by this conjecture, the Steiner tree packing problem has obtained wide attention, and many results have been worked out; see [62, 64–66, 69, 120, 137]. In [90] we set up the relationship between the Steiner tree packing problem and the generalized edge-connectivity. The generalized edge-connectivity and the Steiner tree packing problem have applications in VLSI circuit design; see [51, 52, 125]. In this application, a Steiner tree is needed to share an electronic signal by a set of terminal nodes. Steiner tree is also used in computer communication networks (see [39]) and optical wireless communication networks (see [33]). Another application, which is our primary focus, arises in the Internet domain. Imagine that a given graph G represents a network. We arbitrarily choose k vertices as nodes. Suppose one of the nodes in G is a broadcaster and all other nodes are either users or routers (also called switches). The broadcaster wants to broadcast as many streams of movies as possible, so that the users have the maximum number of choices. Each stream of movie is

1.5 Generalized Edge-Connectivity, Edge Toughness, Strength, and Arboricity

11

broadcasted via a tree connecting all the users and the broadcaster. So in essence we need to find the maximum number Steiner trees connecting all the users and the broadcaster, namely, we want to get .S/, where S is the set of the k nodes. Clearly, it is a Steiner tree packing problem. Furthermore, if we want to know whether for any k nodes, the network G has the above properties, then we need to compute k .G/ D minf.S/g in order to prescribe the reliability and the security of the network.

1.5 Generalized Edge-Connectivity, Edge Toughness, Strength, and Arboricity Toughness of graphs was first studied by Chvátal [34], where he defined the toughness of a graph G as   jSj t.G/ D min W S  V.G/; !.G  S/ > 1 : !.G  S/ We refer to Bauer, Broersma, and Schmeichel [8] for details on the toughness. However, if one defines the edge toughness of G analogously as   jXj  .G/ D min W X  E.G/; !.G  X/ > 1 ; !.G  X/ then it is not difficult to see that .G/ D .G/=2, where .G/ denotes the edgeconnectivity of G. Hence, it is not of much interest. Theorem 1.4.1, due to Nash-Williams and Tutte, can be stated in the following way. Theorem 1.5.1 (Nash-Williams [107], Tutte [133]). A graph G has ` edgedisjoint spanning trees if and only if jXj  `.!.G  X/  1/ for each X  E.G/. Motivated by this result, Peng, Chen, and Koh introduced the following definition of edge toughness 1 .G/ of a graph G:   jXj 1 .G/ D min W X  E.G/; !.G  X/ > 1 : !.G  X/  1 As an immediate consequence of the definition and Theorem 1.5.1, we have the following result. Corollary 1.5.2 ([117]). A graph G has ` edge-disjoint spanning trees if and only if 1 .G/  `. From the definition of 1 .G/, k .G/, and k .G/, we have n .G/ D n .G/ D b1 .G/c:

12

1 Introduction

The strength of a graph G is defined as .G/ D min

XE.G/

jXj ; !.G  X/  !.G/

where the minimum is taken over whenever the denominator is nonzero and !.G/ denotes the number of components of G. From Nash-Williams-Tutte theorem, a multigraph G contains a system of k edgedisjoint spanning trees if and only if for any X  E.G/, jXj  k.!.G  X/  1/. One can see that the concept of the strength of a graph may be derived from the Nash-Williams-Tutte theorem for connected graphs. From the definition of .G/, k .G/, k .G/, and Nash-Williams-Tutte theorem, we have n .G/ D n .G/ D b .G/c for a connected graph G. For more details, we refer to [26, 54, 135]. In addition, the generalized (edge)-connectivity and the strength of a graph can be used to measure the reliability and the security of a network; see [35, 105]. Another closely related graph parameter that has been widely studied is the arboricity of a graph, i.e., the minimum number of subforests needed to cover all of its edges. The arboricity of a graph G is the smallest number of forests whose union is G. A trivial lower bound on the arboricity of a graph with n vertices is the number of edges divided by n  1, since we cannot do better than covering all the edges with a set of edge-disjoint spanning trees. The next theorem is due to Nash-Williams. Theorem 1.5.3 ([108]). Let G be a graph. Then the arboricity of G is  max

 jE.H/j ; jV.G/j  1

where the maximum is taken over all nontrivial subgraphs H of G.

1.6 Generalized Connectivity and Rainbow Indices Last but not the least, we mention the relationship between the generalized connectivity and the rainbow indices of graphs. Actually, we learned the concept of the generalized connectivity from a paper on rainbow indices. The concepts of rainbow indices were introduced by Chartrand et al. in [28]. The definitions are stated as follows. Let G be an edge-colored graph of order n, which means that there is a mapping C W E.G/ ! N where N is the set of positive integers. A tree T in G is called a rainbow tree if no two edges of T have the same color. Let k be a fixed integer with 2  k  n. The graph G is called k-rainbow connected if for

1.6 Generalized Connectivity and Rainbow Indices

13

every set S of k vertices of G, there is a rainbow tree in G connecting the vertices of S, which is called a rainbow S-tree. For a connected graph G, the k-rainbow index rxk .G/ of G is the minimum number of colors needed in order to make G k-rainbow connected. For k D 2, the index rx2 .G/ is called the rainbow connection number of G, denoted by rc.G/, which has been studied extensively. For details, we refer to the book [95] and a survey paper [94]. Furthermore, let k  2 and ` be two integers with 1  `  k .G/. Then, for a connected graph G, the .k; `/-rainbow index rxk;` .G/ is the smallest number of colors needed in an edge coloring of G such that for every set S of k vertices of G, there is ` internally disjoint rainbow S-tree in G. As one can see, if ` is larger than k .G/, this index makes no sense. The definition must be set up on the basis of the generalized connectivity. For ` D 1, the .k; 1/-rainbow index rxk;1 .G/ is reduced to the k-rainbow index rxk ; for k D 2, the .2; `/-rainbow index is called the rainbow `-connectivity rc` , and the .2; 1/-rainbow index is just the rainbow connection number rc. For more about the rainbow indices, we refer to [94, 95] and [21–24, 32, 92, 93].

Chapter 2

Results for Some Graph Classes

In this chapter, we study the generalized k-connectivity for some special graph classes. As one will see, even for special graphs, it is not easy to get the exact values of the generalized connectivity for them. The following two observations are easily seen. Observation 2.1. Let n; k be two integers with 2  k  n. If G is a connected graph of order n, then k .G/  k .G/  ı.G/. Observation 2.2. Let n; k be two integers with 2  k  n. If H is a spanning subgraph of a graph G with order n, then k .H/  k .G/ and k .H/  k .G/.

2.1 Results for the Spanning-Tree Packing Number At first we look at the case for k D n, i.e., the spanning-tree packing numbers of graphs. In his survey paper [114], Palmer listed the spanning-tree packing numbers for some graph classes. Since then, some new results have been derived. We now give an update for Palmer’s table; see Table 2.1. First of all, we give the definitions for the special graph classes. An r-dimensional torus is the Cartesian product of r cycles of sizes at least 3. An r-dimensional torusconnected k-ary hypercube is a special case of an r-dimensional torus, in which the r cycles are all of the same size k, where k  3. Simply, we call a 2-dimensional torus a torus. A graph G is called maximal planar if there is an embedding of G such that each region is bounded by three edges. A maximal toroidal graph is defined in a similar way. A toroidal graph is defined as a graph that can be embedded on the torus but not on the plane. A graph G is called maximal toroidal if G is toroidal and, for some embedding, every region has three sides. Let q be a prime power congruent to 1 (mod 4). The Paley graph Pq has the q elements of the field Fq as its vertices and x are adjacent to y if x  y is a square in Fq . These graphs are regular of degree © The Author(s) 2016 X. Li, Y. Mao, Generalized Connectivity of Graphs, SpringerBriefs in Mathematics, DOI 10.1007/978-3-319-33828-6_2

15

16

2 Results for Some Graph Classes Table 2.1 Results for STP number Graphs or networks Kn .n  1/ Ka;b .1  a  b/ Complete multipartite graph (order n and size m) k-tree (order n and size m) Qn .n  1/ Paley Pq Octahedron Icosahedron Maximal planar Connected cubic (n  6) Connected 4-regular (n  7) Maximal toroidal (n  7) Random r-regular Harary Hn;d Quasi-random r-d torus r-d torus-connected k-ary hypercube r-d hypercube r-d mesh (having 3m dimensions of size at least 3) r-d mesh-connected k-ary hypercube r-d generalized hypercube (mj , size of the j-th clique ; f , number of even-size cliques) r-d mesh-connected tree (h, height of the trees) r-d hyper Petersen network Kn Cm .Ka ı bK1 /Cp .Ka ı bK1 /Kp .Ka ı bK1 /.Kp ı qK1 /

STP number bn=2c bab=.a C b  1/c bm=.n  1/c bm=.n  1/c bn=2c .q  1/=4 2 2 2 1 1 or 2 3 br=2c bb2d=nc=2c .1 C o.1//n=4 r (maximum) r (maximum) br=2c (maximum) 2m C b.r  3m/=2c (maximum) b2r=3c (maximum for k D 3) Pn

jD1 bmj =2c

 df =2e (maximum)

b2r=3c (maximum for h D 2) br=2c (maximum) b.nm.n C 1//=2.nm  1/c b.abp.ab  b C 2//=2.abp  1/c b.abp.ab C p  b  1//=2.abp  1/c b.abpq.ab C pq  b  q//=2.abpq  1/c

.q  1/=2 and hence have diameter 2. Let k be a positive integer. Then a k-tree is a graph defined recursively as follows: the smallest k-tree is the complete graph with k vertices, and a k-tree with n C 1 vertices where n  k is obtained by adding a new vertex adjacent to each of the k arbitrarily selected but mutually adjacent vertices of a k-tree with n vertices. The Harary graph Hn;d is constructed by arranging the n vertices in a circular order and spreading the d edges around the boundary in a nice way, keeping the

2.1 Results for the Spanning-Tree Packing Number

17

chords as short as possible. Harary graph Hn;d is a d-connected graph on n vertices that has exactly b dn c edges, and the structure of Hn;d depends on the parities of d 2 and n: Case 1: d even. Let d D 2r. Then Hn;2r is constructed as follows. It has vertices 0; 1;    ; n  1, and two vertices i and j are jointed if i  r  j  i C r (where addition is taken modulo n). Case 2: d odd, n even. Let d D 2r C 1. Then Hn;2rC1 is constructed by first drawing Hn;2r and then adding edges joining vertex i to vertex iC n2 for 1  i  n2 . Case 3: d odd and n even. Let d D 2r C 1. Then Hn;2rC1 is constructed by first drawing Hn;2r and then adding edges joining vertex 0 to vertices n1 and nC1 and 2 2 nC1 n1 i to vertex i C 2 for 1  i  2 . An r-dimensional mesh is the Cartesian product of r linear arrays. An r-dimensional hypercube is a special case of an r-dimensional mesh, in which the r linear arrays are all of size 2. An r-dimensional mesh-connected k-ary hypercube is a special case of an r-dimensional mesh, in which the r linear arrays are all of the same size k, where k  3. An r-dimensional generalized hypercube is the Cartesian product of r cliques. An r-dimensional mesh-connected tree is the Cartesian product of r complete binary trees of the same height h  2. An r-dimensional hyper Petersen network HPr is the Cartesian product of the well-known Petersen graph and Qr3 , where r  3 and Qn3 denotes an .r  3/-dimensional hypercube. Let Fn be a family of graphs of order n with vertex set Œn D f1; 2; : : : ; ng and some specified probability distribution. The family is quasi-random if with probability N D jSjjSN j C o.n2 / for every subset approaching 1 as n ! 1, we have jEG ŒS; Sj 2 S  Œn. Peng, Chen, and Koh [116] obtained the following bounds of n .G/. Theorem 2.1.1 ([116]). Let G be a graph of order n and edge-connectivity . Then n b 2.n1/ c  n .G/  . The following result is an immediate consequence of Theorem 2.1.1. Theorem2.1.2 ([119]). Let G be a graph of order n and edge-connectivity . Then ˘ n n .G/ D 2.n1/ if and only if G is -regular. Corollary 2.1.3 ([119]).

 nr ˘ (1) If G is a r-regular graph of order n and r  bn=2c, then n .G/ D 2.n1/ . If  nr.n1/ ˘ . r D .n  1/=2, then n .G/ D n .G/ D 2 (2) If G is a vertex-transitive graph or a regular edge-transitive graph of order n m and size m, then n .G/ D b n1 c. Corollary 2.1.4 ([119]). Let Gi .i D 1; 2/ be a connected graph of order ni , size mi , and edge-connectivity i . If Gi is i -regular, then

18

2 Results for Some Graph Classes

n .G1 G2 / D

jm n C m n k 1 2 2 1 : n1 n2  1

We first state the following result, which will play a key role for studying the spanning-tree packing number. Theorem 2.1.5 ([119]). Let G be a graph. Then jE.G/j=.jV.G/j1/  jXj=.!.G X/  1/ for every edge cutset X of G if and only if jE.H/j jE.G/j  jV.H/j  1 jV.G/j  1 for every subgraph H of G. Proof. Let n and m, respectively, be the order and size of G. Suppose jE.G/j=.jV.G/j  1/  jXj=.!.G  X//  1/ for every edge cutset X of G. Let H be any subgraph with n0 vertices and m0 edges. If n0 D n, then m0 =.n0  1/  m=.n  1/. Otherwise, let x be the cardinality of the edge cutset X D E.G/  E.H/, which separates G into H and some isolated vertices. Then m D m0 C x and !.G  X/  n  n0 C 1. Therefore, we have x m  ; n1 n  n0 i.e., m.n  n0 /  .m  m0 /.n  1/. Thus m m0  ; 0 n 1 n1 as required. Conversely, let X be any edge cutset separating G into t components H1 ; H2 ; : : : ; Ht . Then for each component Hi , we have m jE.Hi /j  : jV.Hi /j  1 n1 Thus mD

t X

jE.Hi /j C jXj 

iD1

t X iD1

m m .jV.Hi /j  1/ C jXj D .n  t/ C jXj: n1 n1

Therefore m jXj  ; n1 t1 as required.

t u

2.1 Results for the Spanning-Tree Packing Number

19

Corollary 2.1.6 ([119]). Let G be a connected graph of order n and size m. Then m m n .G/ D b n1 c if and only if jE.H/j  b n1 c.jV.H/j  1/ for every subgraph H of G. The edge toughness of a complete r-partite graph has been determined by Peng, Chen, and Koh [116]. Peng and Tay [119] later gave a simpler proof to the result. Theorem 2.1.7 ([119]). If G is a complete r-partite graph with n vertices and m m edges, then n .G/ D b n1 c. We shall use Theorem 2.1.5 to prove Theorem 2.1.7. To do so, we first establish the following lemma, from which Theorem 2.1.7 follows as an easy consequence. Lemma 2.1.8 ([119]). Let H D Km1 ;m2 ;:::;mr be a complete r-partite graph where r  2 and mr  mr1  : : :  m1  1. If G is a complete r-partite or .r C 1/partite graph obtained from H by adding one vertex, then jE.H/j jE.G/j  : jV.H/j  1 jV.G/j  1 Proof. Note that jE.H/j D jV.H/j C 1. Then jE.H/j jV.H/j  1



P 1i b  a C 2 and a  b C k is even, then k .Ka;b / D

a C b  k j .a  b C k/.b  a C k/ k C : 2 4.k  1/

If k  b  a C 2, then k .Ka;b / D a. The proof of Theorem 2.2.3 is long and complicated, which uses the following seven lemmas, Lemmas 2.2.4 through 2.2.10. The proofs of them are omitted. Note that k .G/ D minf.S/g, where the minimum is taken over all k-element subsets S of V.G/. Let X D fx1 ; x2 ; : : : ; xa g and Y D fy1 ; y2 ; : : : ; yb g be the bipartition of Ka;b . Actually, all vertices in X are equivalent and all vertices in Y are equivalent. So instead of considering all k-element subsets S of V.G/, we can restrict our attention to the k-element subsets Si D fx1 ; x2 ; : : : ; xi ; y1 ; y2 ; : : : ; yki g for 0  i  k. Notice that if i > a or k  i > b, then Si does not exist. So we need only to consider Si for maxf0; k  bg  i  minfa; kg. Now, let T be a maximum set of internally disjoint Si -Steiner trees. Let T0 be the set of Si -Steiner trees whose vertex set is Si ; let T1 be the set of Si -Steiner trees whose vertex set is Si [ fug, where u … Si ; and let T2 be the set of Si -Steiner trees whose vertex set is Si [ fu; vg, where u; v … Si and they belong to different partitions.

22

2 Results for Some Graph Classes

Lemma 2.2.4 ([79]). Let T be a maximum set of internally disjoint Si -Steiner trees. Then we can always find a set T 0 of internally disjoint Si -Steiner trees such that jT j D jT 0 j and T 0  T0 [ T1 [ T2 . We can assume that the maximum set T of internally disjoint Si -Steiner trees is contained in T0 [ T1 [ T2 . We will define the standard structure of Steiner trees in T0 , T1 , and T2 , respectively. Every tree in T0 is of standard structure. A Steiner tree T in T1 with vertex set V.T/ D Si [ fug, where u 2 X  Si , is of standard structure, if u is adjacent to every vertex in Si \ Y. Since jE.T/j D jV.T/j  1 D k and dT .u/ D jSi \ Yj D k  i, there remain i edges incident with Si \ X. We know that jSi \ Xj D i and each vertex must have degree at least 1 in T. So every vertex in Si \ X has degree 1. A Steiner tree T in T1 with vertex set V.T/ D Si [ fvg, where v 2 Y  Si , is of standard structure, if v is adjacent to every vertex in Si \ X. Similarly, every vertex in Si \ Y has degree 1. A Steiner tree T in T2 with vertex set V.T/ D Si [ fu; vg, where u 2 X  Si and v 2 Y  Si , is of standard structure, if u is adjacent to every vertex in Si \ Y, v is adjacent to every vertex in Si \ X, and u is adjacent to v. We then denote the resulting tree T by Tu;v . Denote the set of trees in T0 , T1 , and T2 with the standard structure by T0 , T1 , and T2 , respectively. Clearly, T0 D T 0 . Lemma 2.2.5 ([79]). Let T be a maximum set of internally disjoint Si -Steiner trees, T  T0 [ T1 [ T2 . Then we can always find a set T 00 of internally disjoint Si -Steiner trees, such that jT j D jT 00 j and T 00  T0 [ T1 [ T2 . So, we can assume that the maximum set T of internally disjoint Si -Steiner trees is contained in T0 [T1 [T2 . Namely, all Steiner trees in T are of standard structure. For simplicity, we denote the union of the vertex sets of all Steiner trees in set T by V.T / and the union of the edge sets of all Steiner trees in set T by E.T /. Let T0 WD T \ T0 , T1 WD T \ T1 , and T2 WD T \ T2 . Then T D T0 [ T1 [ T2 . Lemma 2.2.6. Let T  T0 [ T1 [ T2 be a maximum set of internally disjoint Si -Steiner trees. Then either X  V.T / or Y  V.T /. We conclude that if T is a maximum set of internally disjoint Si -Steiner trees, then X  V.T / or Y  V.T /. Lemma 2.2.7 ([79]). Let T  T0 [T1 [T2 be a maximum set of internally disjoint Si -Steiner trees and T D T0 [ T1 [ T2 . If there is a vertex x 2 X  V.T / and a tree T 2 T1 with vertex set Si [ fyg, where y 2 Y  Si , then we can find a maximum set T 0 D T00 [ T10 [ T20 of internally disjoint Si -Steiner trees, such that T00 D T0 , jT10 j D jT1 j  1, and jT20 j D jT2 j C 1. The case that there is a vertex y 2 Y  V.T / and a tree T 2 T1 with vertex set Si [ fxg, where x 2 X  Si , is similar. We will show that we can always find a maximum set T of internally disjoint Si -Steiner trees such that all vertices in V.T1 /  Si belong to the same partition. Lemma 2.2.8 ([79]). Let p, q be two nonnegative integers. If p.k1/Cqi  i.ki/ and there are q vertices u1 ; u2 ; : : : ; uq 2 X  Si , then we can always find p Steiner

2.2 Results for the Generalized k-Connectivity

23

trees T1 ; T2 ; : : : ; Tp in T0 and q Steiner trees TpC1 ; TpC2 ; : : : ; TpCq in T1 , such that V.Tj / D Si for 1  j  p, V.TpCm / D Si [ fum g for 1  m  q, and Tr ; Ts are edge disjoint for 1  r < s  p C q. Similarly, if p.k  1/ C q.k  i/  i.k  i/, and there are q vertices v1 ; v2 ; : : : ; vq 2 Y  Si , then we can always find p Steiner trees T1 ; T2 ; : : : ; Tp in T0 and q Steiner trees TpC1 ; TpC2 ; : : : ; TpCq in T1 , such that V.Tj / D Si for 1  j  p, V.TpCm / D Si [ fvm g for 1  m  q, and Tr ; Ts are edge disjoint for 1  r < s  p C q. Lemma 2.2.9 ([79]). Let T  T0 [T1 [T2 be a maximum set of internally disjoint Si -Steiner trees and T D T0 [ T1 [ T2 . If there are s Steiner trees T1 ; T2 ; : : : ; Ts 2 T1 with vertex set Si [ fu1 g, Si [ fu2 g, : : :, Si [ fus g, respectively, where uj 2 X  Si for 1  j  s, and t Steiner trees TsC1 ; TsC2 ; : : : ; TsCt 2 T1 with vertex set Si [ fv1 g, Si [ fv2 g, : : :, Si [ fvt g, respectively, where vj 2 Y  Si for 1  j  t, then we can find a set T 0 D T00 [ T10 [ T20 of internally disjoint Si -Steiner trees such that jT j D jT 0 j and all vertices in V.T10 /  Si belong to the same partition. From Lemmas 2.2.7 and 2.2.9, if T 0 is a set of internally disjoint Si -Steiner trees which we find currently, X  V.T / ¤ ; and Y  V.T / ¤ ;, then no matter how many edges there are in E.Ka;b ŒSi / n E.T 0 /, we always add to T 0 the trees in T2 rather than the trees in T1 to form a larger set of internally disjoint Si -Steiner trees. Lemma 2.2.10 ([79]). Let T  T0 [ T1 [ T2 be a maximum set of internally disjoint Si -Steiner trees and T D T0 [ T1 [ T2 . If V.T /  V.G/ and T0 ¤ ;, then we can find a maximum set T 0 D T00 [T10 [T20 of internally disjoint Si -Steiner trees such that jT00 j D jT0 j  1, jT10 j D jT1 j C 1, and T20 D T2 . We can assume that for the maximum set T of internally disjoint Si -Steiner trees, either V.T / D V.G/ or T0 D ;. Moreover, if T 0 is a set of internally disjoint Si -Steiner trees which we find currently, V.T 0 /  V.G/, and the edges in E.Ka;b ŒSi / n E.T 0 / can form a tree T in T0 , then we will add the tree T 00 into T 0 in Lemma 2.2.10 rather than the tree T to form a larger set of internally disjoint Si -Steiner trees. Using the above seven lemmas, the proof of Theorem 2.2.3 then goes as follows: Proof of Theorem 2.2.3. Let X D fx1 ; x2 ; : : : ; xa g and Y D fy1 ; y2 ; : : : ; yb g be the bipartition of Ka;b . As we have mentioned, we can restrict our attention to the k-element subsets Si D fx1 ; x2 ; : : : ; xi ; y1 ; y2 ; : : : ; yki g for maxf0; k  bg  i  minfa; kg. From the above lemmas, we can determine our rule to find the maximum set of internally disjoint Steiner trees connecting Si . Namely, we first find as many Steiner trees in T2 as possible, next we find as many Steiner trees in T1 as possible, and finally we find as many Steiner trees in T0 as possible. Let T be the maximum set of internally disjoint Si -Steiner trees we finally find. We now compute jT j: Case 1. k  b  a C 2. Note that .S0 / D a. For S1 , since k  b  a C 2, it follows that b  .k  1/ D b  k C 1  a  2 C 1 D a  1, and hence jT2 j D a  1. If b  k C 1 D a  1, then

24

2 Results for Some Graph Classes

jT1 j D 0 and jT0 j D 1. If b  k C 1 > a  1, then jT1 j D 1 and jT0 j D 0. No matter which case happens, we have .S1 / D jT2 j C jT1 j C jT0 j D a. For Si .i  2/, since k  b  a C 2, it follows that b  .k  i/ D b  k C i  a  2 C i > a  i, and hence jT2 j D a  i. Since b  k C i  .a  i/ D b  a  k C 2i  2 C 2i  i, it follows that jT1 j D i and jT0 j D 0. Thus .Si / D jT2 j C jT1 j C jT0 j D a. In summary, if k  b  a C 2, then k .G/ D a. Case 2. k > b  a C 2. First, let us compare .Si / with .Ski / for 0  i  b 2k c. If a D b, then .Si / D .Ski /. So we may assume that a < b. For i D 0, .S0 / D a < b D .Sk /. For 1  i  b 2k c, we will give the expressions of .Si / and .Ski /. For Si , since every pair of vertices u 2 X  Si and v 2 Y  Si can form a tree Tu;v , then jT2 j D minfa  i; b  .k  i/g, and hence ( jT2 j D

ai

if i 

b  k C i if i
0. First, we consider the S case 4 > 0. Then 2  2  1 D 4 . For each edge ei incident with w not in P1 , i 2 f1; : : : ; 4 g, we choose a path Pi 2 P2 such that Pi ¤ Pj whenever i ¤ j; let Ti be the tree formed by Pi , w, and ei . Then fTi j 1  i  4 g [ fTP j P 2 P3 g is a system of 3 C 4 edge-disjoint Steiner trees. The graph G obtained from G by removing these trees contains a subdivision of a 21 -edge-connected graph on fu; v; wg. Since 21 D 3 4  b 8` 3 4 c  6 8.`3 4 /C3  b c, it follows that G contains `  3  4 edge-disjoint S-Steiner trees, 6 and hence G contains ` edge-disjoint S-Steiner trees.

54

4 Sharp Bounds of the Generalized (Edge-)Connectivity

Next, we consider the case 4 D 0. Choose an arbitrary subsystem R of P3 consisting of 03 D 3  1 C 2 D   21  0 paths. The graph G obtained from G by removing the trees TR .R 2 R/ contains 1 paths of P2 [P3 ; so G contains a 8.`03 /C3 subdivision of a 21 -edge-connected graph on S. Since 21 D 03  b c, 6 it follows that G contains `03 edge-disjoint S-Steiner trees, and hence G contains ` edge-disjoint S-Steiner trees. t u Although the graphs considered in Theorem 4.2.1 may contain multiple edges, it holds true also for simple graphs. In the following we are concerned with the generalized edge-connectivity only for simple graphs. By Theorem 4.2.1, Li, Mao, and Sun derived a sharp lower bound of 3 .G/ and gave graphs attaining the bound. Proposition 4.2.4 ([90]). Let G be a connected graph with n vertices. For every two integers ` and r with `  0 and r 2 f0; 1; 2; 3g, if .G/ D 4` C r, then 3 .G/  3` C dr=2e. Moreover, the lower bound is sharp. We simply write 3 .G/  32 . 4 Proof. Let  D b 8tC3 c. From Theorem 4.2.1, we have 3 .G/  t. If  D 4`, 6 8tC3 8tC3 then 4` < 6 , since 6 is not an integer. Thus 3 .G/  t > 3`  38 , and hence 3 .G/  3`. With a similar method, we can obtain that 3 .G/  3`C1 if  D 4`C1 and 3 .G/  3` C 2 if  D 4` C 3. Note that there exists no integer t such that 4` C 2 D b 8tC3 c if  D 4` C 2. But 6 a graph G with .G/ D 4` C 2 is also .4` C 1/-edge-connected, and so we have 3 .G/  3` C 1. If  D 4`, then 3 .G/  3`. If  D 4` C 3, then 3 .G/  3` C 2. If  D 4` C 1 or  D 4` C 2, then 3 .G/  3` C 1. So the result holds. Simply, we write 3 .G/  32 . t u 4 They gave the following graph class to show that the lower bound is sharp. Example 4.2. For  D 4` with `  1, let P D X1 [ X2 and Q D Y1 [ Y2 be two cliques with jX1 j D jY1 j D 2` and jX2 j D jY2 j D 2`. Let x; y be adjacent to every vertex in P; Q, respectively, and z be adjacent to every vertex in X1 and Y1 . Finally, we finish the construction of G by adding a perfect matching between X2 and Y2 . It can be checked that  D 4` and .S/  3`. One can also check that for any other set of three vertices of G, the number of edge-disjoint Steiner trees connecting them is not less than 3`. So 3 .G/ D 3` and the graph G attains the lower bound (Fig. 4.9). For  D 4` C 1, let jX1 j D jY1 j D 2` C 1 and jX2 j D jY2 j D 2`; for  D 4` C 2, let jX1 j D jY1 j D 2` C 1 and jX2 j D jY2 j D 2` C 1; for  D 4` C 3, let jX1 j D jY1 j D 2` C 2 and jX2 j D jY2 j D 2` C 1, where `  1. Similarly, one Fig. 4.9 The graph with .G/ D 4` and 3 .G/ D 3` that attains the lower bound of Proposition 4.2.4

P

Q

X1

Y1 y

z

x

X2

Y2

4.3 Bounds for Planar and Line Graphs

55

can check that 3 .G/ D 3` C 1 for  D 4` C 1, 3 .G/ D 3` C 1 for  D 4` C 2, and 3 .G/ D 3` C 2 for  D 4` C 3. For the case ` D 0, G D Pn satisfies that .G/ D 3 .G/ D 1; G D Cn satisfies that .G/ D 2 and 3 .G/ D 1; and G D Ht satisfies that .G/ D 3 and 3 .G/ D 2, where Ht denotes the graph obtained from t copies of K4 , say K41 ; K42 ; : : : ; K4t , by identifying a vertex ui of K4i and a vertex viC1 of K4iC1 , where uiC1 ¤ viC1 and 1  i  t  1. Li and Mao [86] derived a lower bound from Corollary 1.4.3. Proposition 4.2.5 ([86]). For a connected graph G of order n and an integer k with 3  k  n, we have k .G/  b 12 .G/c. Moreover, the lower bound is sharp. In order to show the sharpness of this lower bound for k D n, they showed that the Harary graph Hn;2r attains this bound. For a general k .3  k  n/, one can check that the cycle Cn can attain the lower bound since 12 .Cn / D 1 D k .Cn /. Li, Mao, and Sun gave a sharp upper bound of k .G/. Proposition 4.2.6 ([90]). For any graph G of order n, k .G/  .G/. Moreover, the upper bound is sharp. Li and Mao [86] showed that the monotone property of k is true for 2  k  n although it is not true for k . Proposition 4.2.7 ([86]). For two integers k and n with 2  k  n  1 and a connected graph G, we have kC1 .G/  k .G/. From Observation 2.1, we know that k .G/  k .G/  ı. Actually, Li, Mao, and Sun [90] showed that the graph G D Kk _ .n  k/K1 .n  3k/ satisfies that k .G/ D k .G/ D .G/ D .G/ D ı.G/ D k, which implies that the upper bounds of Observation 2.1, Proposition 4.2.6, and Theorem 4.1.7 are sharp. Li and Mao [89] gave a sufficient condition for k .G/  ı  1. Proposition 4.2.8 ([89]). Let G be a connected graph of order n with minimum degree ı. If there are two adjacent vertices of degree ı, then k .G/  ı  1 for 3  k  n. Moreover, the upper bound is sharp.

4.3 Bounds for Planar and Line Graphs With the above bounds, we will focus on their applications in this section. From Theorems 4.1.1 and 4.1.7, Li, Li, and Zhou derived sharp bounds for planar graphs. Theorem 4.3.1 ([83]). If G is a connected planar graph, then .G/  1  3 .G/  .G/. Motivated by constructing graphs to show that the upper and lower bounds are sharp, they obtained some lemmas. By the well-known Kuratowski’s theorem [20], they verified the following lemma.

56

4 Sharp Bounds of the Generalized (Edge-)Connectivity

Lemma 4.3.2 ([83]). For a connected planar graph G with 3 .G/ D k, there are no 3 vertices of degree k in G, where k  3. They also studied the generalized 3-connectivity of four kinds of graphs. Lemma 4.3.3 ([83]). If .G/  3, then 3 .G  e/  2 for any edge e 2 E.G/. An `-connected graph G is called minimally `-connected if the graph G n e is not `-connected for any edge e of G. Lemma 4.3.4 ([83]). If G is a planar minimally 3-connected graph, then 3 .G/ D 2. Lemma 4.3.5 ([83]). Let G be a 4-connected graph, and let H be a graph obtained from G by adding a new vertex w and joining it to three vertices of G. Then 3 .H/ D .H/ D 3. Lemma 4.3.6 ([83]). If G is a planar minimally 4-connected graph, then 3 .G/ D 3. If G is a connected planar graph, then 1  .G/  5. By Theorem 4.3.1, we have .G/  1  3 .G/  .G/. Then, for each 1  .G/  5, they gave some classes of planar graphs attaining the bounds of 3 .G/, respectively: Case 1. .G/ D 1. For any graph G with .G/ D 1, obviously 3 .G/  1 and so 3 .G/ D .G/ D 1. Therefore, all planar graphs with connectivity 1 can attain the upper bound, but cannot attain the lower bound. Case 2. .G/ D 2. Let G be a planar graph with .G/ D 2 and having two adjacent vertices of degree 2. Then by Proposition 4.1.10, 3 .G/  1 and so 3 .G/ D 1 D .G/  1. Therefore, this graph class attains the lower bound. For example, for any cycle C, we have 3 .C/ D 1 and .C/ D 2. Let G be a planar minimally 3-connected graph. By the definition, for any edge e 2 E.G/, 3 .G n e/ D 2. From Lemma 4.3.3 and Theorem 4.1.7, we have 3 .G n e/ D 2. Therefore, the 2-connected planar graph G n e attains the upper bound. Case 3. .G/ D 3. For any planar minimally 3-connected graph G, we know that .G/ D 3, and by Lemma 4.3.4, 3 .G/ D 2 D .G/  1. So this graph class attains the lower bound. Let G be a planar 4-connected graph, and let H be a graph obtained from G by adding a new vertex w in the interior of a face for some planar embedding of G and joining it to 3 vertices on the boundary of the face. Then H is still planar, and by Lemma 4.3.5, one can immediately get that 3 .H/ D .H/ D 3, which means that H attains the upper bound. Case 4. .G/ D 4. For any planar minimally 4-connected graph G, one knows that .G/ D 4, and by Lemma 4.3.6, 3 .G/ D 3 D .G/  1. So this graph class attains the lower bound. For every graph in Fig. 4.10, the vertex in the center has degree 4, and it can be checked that for any two vertices, there always exist four internally disjoint paths connecting them, which means that .G/ D 4. It can also be checked that for any 3 vertices there always exist four internally disjoint trees connecting them. Combining this with Theorem 4.1.7, one can get

4.3 Bounds for Planar and Line Graphs

57

Fig. 4.10 Graphs for the upper bound of case 4

that 3 .G/ D 4. Therefore, the graphs attain the upper bound. Moreover, we can construct a series of graphs according to the pattern of Fig. 4.10, which attain the upper bound. Case 5. .G/ D 5. For any planar graph G with .G/ D 5, one can get that 3 .G/ D 4. So, any planar graph G with connectivity 5 can attain the lower bound, but obviously cannot attain the upper bound. Similarly, the following result is obvious from Propositions 4.2.4 and 4.2.6. Proposition 4.3.7 ([90]). If G be a connected planar graph, then .G/  1  3 .G/  .G/. Chartrand and Stewart [29] investigated the relation between the connectivity and edge-connectivity of a graph and its line graph. Theorem 4.3.8 ([29]). If G is a connected graph, then: (1) .L.G//  .G/ if .G/  2. (2) .L.G//  2.G/  2. (3) .L.L.G///  2.G/  2. By Proposition 4.2.4, Li, Mao, and Sun also considered the generalized 3-connectivity and 3-edge-connectivity for line graphs. Proposition 4.3.9 ([90]). If G is a connected graph, then: (1) 3 .G/  3 .L.G//. (2) 3 .L.G//  32 3 .G/  2. (3) 3 .L.L.G//  32 3 .G/  2. Let L0 .G/ D G and L1 .G/ D L.G/. Then for r  2, the r-th iterated line graph L .G/ of G is defined by L.Lr1 .G//. The next statement follows immediately from Proposition 4.3.9 and a routine application of induction. r

Corollary 4.3.10 ([90]). If G is a connected graph, then 3 .Lr .G//  . 32 /r .3 .G/ 4/ C 4, and 3 .Lr .G//  . 32 /br=2c .3 .G/  4/ C 4.

Chapter 5

Graphs with Given Generalized Connectivity

From the last chapter, we know that 1  k .G/  n  dk=2e and 1  k .G/  n  dk=2e for a connected graph G of order n. In this chapter, we characterize the graphs with k .G/ D ndk=2e; ndk=2e1, and k .G/ D ndk=2e; ndk=2e1.

5.1 Graphs with the Largest Generalized Connectivity Li, Mao, and Sun [90] characterized graphs attaining the upper bounds, namely, graphs with k .G/ D n  dk=2e or k .G/ D n  dk=2e. Since a complete graph Kn possesses the maximum generalized (edge-)connectivity, they wanted to find out the critical value of the number of edges, denoted by t, such that the generalized (edge-) connectivity of the resulting graph will keep being n  dk=2e by deleting t edges from a complete graph Kn but will not keep being n  dk=2e by deleting t C 1 edges. By further investigation, they conjectured that t may be 0 for k even and k1 2 for k odd. For a connected graph G, we choose S  V.G/ with jSj D k. Let T be a maximum set of edge-disjoint S-Steiner trees in G. Let T1 be the set of S-Steiner trees in T whose edges belong to E.GŒS/ and T2 be the set of S-Steiner trees N where SN D V.G/  S. Thus, T D T1 [ T2 containing at least one edge of EG ŒS; S, (throughout this chapter, T , T1 , T2 are always defined as this). Lemma 5.1.1 ([90]). Let G be a connected graph, S  V.G/, jSj D k, and T be an N if T 2 T2 , S-Steiner tree. If T 2 T1 , then T uses k  1 edges of E.GŒS/ [ EG ŒS; S; N then T uses at least k edges of E.GŒS/ [ EG ŒS; S. Proof. For each Steiner tree T in T1 , T uses k1 edges in E.GŒS/, and hence T uses N For T 2 T2 , by deleting all the vertices of T from k  1 edges of E.GŒS/ [ EG ŒS; S. N we obtain some components of T in S, denoted by C1 ; C2 ;    ; Cr . Let jCi j D ci . S,

© The Author(s) 2016 X. Li, Y. Mao, Generalized Connectivity of Graphs, SpringerBriefs in Mathematics, DOI 10.1007/978-3-319-33828-6_5

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5 Graphs with Given Generalized Connectivity

P Then jE.Ci /j D ci  1 and riD1 .ci  1/ D k  r. Since there exists one edge of N where 1  i  r, it follows that T uses .k  r/ C r D k T between each Ci and S, N edges in E.GŒS/ [ EG ŒS; S. t u Lemma 5.1.2 ([90]). For even k with 4  k  n, k .Kn n e/ < n  e 2 E.Kn /.

k 2

for any

Lemma 5.1.3 ([90]). If k is odd with 3  k  n and M is an edge set of the complete graph Kn such that jMj  kC1 , then k .Kn n M/ < n  kC1 . 2 2 can choose S  V.G/ such that jSj D k and jM \ Proof. Let G D Kn n M. We N  kC1 . Let jT j D y and jT1 j D x. Since each tree T 2 T1 E.Kn ŒS/ [ EKn ŒS; S/j 2  N it follows that jT1 j D x  k =.k  1/ D k1 . uses k  1 edges in E.GŒS/ [ EG ŒS; S, 2 2 N and From Lemma 5.1.1, each tree T 2 T2 uses k edges of E.GŒS/ [ EG ŒS; S, N C jE.GŒS/j, that is, x.k  1/ C .y  x/k  hence 1/ C jT2 jk  jEG ŒS; Sj k jT1 j.k kC1 k1 Ck.nk/ . So  .G/ D y  CnkC kx  kC1  n kC1  2k1 < n kC1 . k 2 2 2 2k 2 2 t u Lemma 5.1.4 ([90]). If n is odd and M is an edge set of the complete graph Kn such that 0  jMj  n1 , then G D Kn n M contains n1 edge-disjoint spanning 2 2 trees. Sp Proof. Let P D iD1 Vi be a partition of V.G/ with jVi j D ni .1  i  p/ and Ep be the set of edges between distinct blocks  The case p D 1  of P Ppin G. is trivial; thus, we assume p  2. Then jEp j  n2  iD1 n2i  jMj  n2    Pp   Pp ni  n1 . We will show that n2  iD1 n2i  n1  n1 .p  1/, that is, iD1 2  2 P 2 2   Pp   p ni n1 n1 .n  p/ 2  iD1 2 . It suffices to prove that .n  p/ 2  maxf iD1 n2i g. Pp ni  Since f .n1 ; n2 ;    ; np / D iD1 2 obtains its maximum value when n1 D n2 D     D np1 D 1 and np D n  p C 1, we need to show the inequality .n  p/ n1 2 npC1 1 p2 .p  1/ C , that is, .n  p/ 2  0. It is easy to see that the inequality 2 2   Pp   .p  1/. From Theorem 1.4.1, holds. Thus, jEp j  n2  iD1 n2i  jMj  n1 2 n1 we know that there exist 2 edge-disjoint spanning trees (note that we can use the result of Theorem 1.4.1, although Nash-Williams and Tutte considered multigraphs, but here we are concerned with the generalized connectivity and generalized edgeconnectivity for simple graphs). t u We want to find k1 edge-disjoint spanning trees in GŒS (by the definition of 2 internally disjoint Steiner trees, these trees are internally disjoint S-Steiner trees, as required). Then the basic idea is to seek for some edges in GŒS and let them together N form n  k internally disjoint S-Steiner trees. with the edges of EG ŒS; S Li, Mao, and Sun [90] proved that there are indeed n  k internally disjoint S-Steiner trees in the premise that GŒS contains k1 edge-disjoint spanning trees. 2 trees. Actually, Lemma 5.1.4 can guarantee the existence of such k1 2 Theorem 5.1.5 ([90]). Let k; n be two integers with 2  k  n. Then for a connected graph G of order n, k .G/ D n  dk=2e if and only if G D Kn for k even; G D Kn n M for k odd, where M is an edge set such that 0  jMj  k1 . 2

5.1 Graphs with the Largest Generalized Connectivity

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Proof. Suppose that k is even. From Theorem 2.2.1, we have k .Kn / D n  2k . Actually, the complete graph Kn is the unique graph with this property. It suffices to show that k .Kn n e/ < n  2k for any e 2 E.Kn /. From Lemma 5.1.2 and Observation 2.1, we know that k .Kn n e/  k .Kn n e/ < n  2k for e 2 E.Kn /. Thus, the result holds for k even. From now on, we assume that k is odd. Let G be a graph of order n such that k .G/ D n  kC1 . Since G is connected, we can consider G as a graph obtained by 2 , deleting some edges from the complete graph Kn . If G D Kn n M with jMj  kC1 2 then it follows from Observation 2.1 and Lemma 5.1.3 that k .Kn n M/  k .Kn n M/ < n  kC1 , a contradiction. Thus, G D Kn n M, where 0  jMj  k1 . 2 2 if G D K n M where 0  Conversely, we will show that k .G/  n  kC1 n 2 kC1 k1 . It suffices to prove that  .G/  n  for jMj D . Let S D jMj  k1 k 2 2 2 N fu1 ; u2 ;    ; uk g  V.G/ and S D fw1 ; w2 ;    ; wnk g. We have the following two cases to consider: Case 1. M  E.Kn ŒS/ [ E.Kn ŒS/. N Then jM 0 j C jM 00 j D jMj D k1 Let M 0 D M \ E.Kn ŒS/ and M 00 D M \ E.Kn ŒS/. 2 k1 0 00 and 0  jM j; jM j  2 . We can consider GŒS as a graph obtained by deleting edgejM 0 j edges from the complete graph Kk . From Lemma 5.1.4, there exist k1 2 k1 disjoint spanning trees in GŒS. Actually, these 2 edge-disjoint trees are all internally disjoint Steiner trees connecting S in GŒS. All these trees together with the trees Ti induced by the edges in fwi u1 ; wi u2 ;    ; wi uk g .1  i  n  k/ form internally disjoint S-Steiner trees, and hence .S/  n  kC1 . n  kC1 2 2 N Case 2. M ª E.Kn ŒS/ [ E.Kn ŒS/. N Let M 0 D M \ E.Kn ŒS/ and In this case, there exist some edges of M in EKn ŒS; S. 00 0 00 N M D M \ E.Kn ŒS/, and let jM j D m1 and jM j D m2 . Clearly, 0  mi  k3 .i D 2 N we let jEKn ŒM Œwi ; Sj D xi , where 1  i  n  k. Without loss 1; 2/. For wi 2 S, P k1 of generality, let x1  x2      xnk . Thus nk and iD1 xi C m1 C m2 D 2 jEG Œwi ; Sj D k  xi . Our basic idea is to seek for some edges in GŒS and let them N form n  k internally disjoint S-Steiner trees. together with the edges of EG ŒS; S N without loss of generality, let S1 D fu1 ; u2 ;    ; ux1 g such that uj w1 2 For w1 2 S, M .1  j  x1 / and S2 D S  S1 D fux1 C1 ; ux1 C2 ;    ; uk g. Clearly, S D S1 [ S2 and uj w1 2 E.G/ .x1 C 1  j  k/, namely, S2 D NG .w1 / \ S. One can see that the tree T10 induced by the edges in fw1 ux1 C1 ; w1 ux1 C2 ;    ; w1 uk g is a Steiner tree connecting S2 . Our idea is to seek for x1 edges in EG ŒS1 ; S2  and add them to T10 to form a Steiner tree connecting S. For each uj 2 S1 .1  j  x1 /, we claim that jEG Œuj ; S2 j  1. Otherwise, let jEG Œuj ; S2 j D 0. Then jEKn ŒM Œuj ; S2 j D k  x1 , and hence jMj  jEKn ŒM Œuj ; S2 j C dKn ŒM .w1 /  .k  x1 / C x1 D k, which . Since jEG Œuj ; S2 j  1 for each uj .1  j  x1 /, we can contradicts to jMj D k1 2 find a vertex ur .x1 C 1  r  k/ such that e1;j D uj ur 2 E.GŒS/. Let M1 D fe1;1 ; e1;2 ;    ; e1;x1 g and G1 D G n M1 . Thus the tree T1 induced by the edges in fw1 ux1 C1 ; w1 ux1 C2 ;    ; w1 uk ; e1;1 ; e1;2 ;    ; e1;x1 g is our desired one.

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N without loss of generality, let S1 D fu1 ; u2 ;    ; ux2 g such that uj w2 2 For w2 2 S, M .1  j  x2 / and S2 D S  S1 D fux2 C1 ; ux2 C2 ;    ; uk g. Clearly, S D S1 [ S2 and uj w2 2 E.G/ .x2 C 1  j  k/, namely, S2 D NG .w2 / \ S. One can see that the tree T20 induced by the edges in fw2 ux2 C1 ; w2 ux2 C2 ;    ; w2 uk g is a Steiner tree connecting S2 . Our idea is to seek for x2 edges in EG1 ŒS1 ; S2  and add them to T20 to form a Steiner tree connecting S. For each uj 2 S1 .1  j  x2 /, we claim that jEG1 Œuj ; S2 j  1. Otherwise, we let jEG1 Œuj ; S2 j D 0. For e … EG1 Œuj ; S2 , e 2 M or e 2 M1 D fe1;1 ; e1;2 ;    ; e1;x1 g. Then jEKn ŒM Œuj ; S2 j  k  x2  x1 , and hence jMj  jEKn ŒM Œuj ; S2 j C dKn ŒM .w1 / C dKn ŒM .w2 /  .k  x2  x1 / C x1 C x2 D k, which contradicts to jMj D k1 . Since jEG1 Œuj ; S2 j  1 for each uj .1  j  x2 /, 2 we can find a vertex ur .x2 C 1  r  k/ such that e2;j D uj ur 2 E.G1 ŒS/. Let M2 D fe2;1 ; e2;2 ;    ; e2;x2 g and G2 D G1 nM2 . Thus the tree T2 induced by the edges in fw2 ux2 C1 ; w2 ux2 C2 ;    ; w2 uk ; e2;1 ; e2;2 ;    ; e2;x2 g is our desired tree. Clearly, T2 and T1 are two internally disjoint S-Steiner trees. For wi 2 SN .3  i  n  k/, without loss of generality, let S1 D fu1 ; u2 ;    ; uxi g such that uj wi 2 M .1  j  xi / and S2 D S  S1 D fuxi C1 ; uxi C2 ;    ; uk g. Clearly, S D S1 [ S2 and wi uj 2 E.G/ .xi C 1  j  k/, namely, S2 D NG .wi / \ S. One can see the tree Ti0 induced by the edges in fwi uxi C1 ; wi uxi C2 ;    ; wi uk g is a Steiner tree connecting S2 . Our idea is to seek for xi edges in EGi1 ŒS1 ; S2  and add them to Ti0 to form a Steiner tree connecting S. For each uj 2 S1 .1  j  xi /, we claim that jEGi1 Œuj ; S2 j  1. Otherwise, let jEGi1 Œuj ; S2 j D 0. For e … EGi1 Œuj ; S2 , S P we have that e 2 M or e 2 i1 M . Then jEKn ŒM Œuj ; S2 j  k  xi  i1 r xr PirD1 r Pi Pi and jMj  jEKn ŒM Œuj ; S2 j C r dKn ŒM .wr /  .k  r xr / C r xr D k, which contradicts to jMj D k1 . Since jEGi1 Œuj ; S2 j  1 for each uj .1  j  xi /, we 2 can find a vertex ur .xi C 1  r  k/ such that ei;j D uj ur 2 E.Gi1 ŒS/. Let Mi D fei;1 ; ei;2 ;    ; ei;xi g and Gi D Gi1 n Mi . Thus the tree Ti induced by the edges in fwi uxi C1 ; wi uxi C2 ;    ; wi uk ; ei;1 ; ei;2 ;    ; ei;xi g is our desired one (note that if xi D 0, then we do not need to search for some edges of E.Gi1 ŒS/, and Ti induced by the edges in fwi u1 ; wi u2 ;    ; wi uk g is our desired tree). Clearly, Ti and Tj .1  j  i  1/ are two internally disjoint S-Steiner trees. We continue this procedure until we find out n  k internally disjoint S-Steiner trees, say T1 ; T2 ;    ; Tnk . Now S we terminate this procedure. Clearly, we 0 can consider Gnk ŒS D GŒS n . nk by deleting jM j C iD1 Mi / as a graph obtained Pnk Pnk jM j edges from the complete graph K . Since x C m1 C m2 D k1 , i k iD1 iD1 i 2 Pnk k1 k1 it follows that 1  iD1 jMi j C m1  2 . From Lemma 5.1.4, there exist 2 edge-disjoint S-Steiner trees in GŒS (note that these trees can be edge disjoint by the definition of generalized k-connectivity). These trees together with T1 ; T2 ;    ; Tnk form n  kC1 internally disjoint S-Steiner trees, and hence .S/  n  kC1 . 2 2 for any S  V.G/ and From the above discussion, we get that .S/  n  kC1 2 . From this together with Proposition 4.1, we jSj D k, and hence k .G/  n  kC1 2 kC1 t u have k .G/ D n  2 . Combining Theorem 5.1.5 and Observation 2.1, they obtained the following theorem for k .G/.

5.2 Graphs with the Second Largest Generalized Connectivity

63

Theorem 5.1.6 ([90]). Let k; n be two integers with 2  k  n. Then for a connected graph G of order n, k .G/ D n  dk=2e if and only if G D Kn for k even; G D Kn n M for k odd, where M is an edge set such that 0  jMj  k1 . 2 Proof. Suppose that k is even. From Proposition 4.2 and Lemma 5.1.2, we have that k .Kn / D n  2k if and only if G D Kn . Suppose that k is odd. If G D Kn n /, then it follows from Observation 2.1 and Theorem 5.1.5 that M .0  jMj  k1 2 . From this together with Proposition 4.2, we know that k .G/  k .G/ D n  kC1 2 . Conversely, assume that k .G/ D n  kC1 . Since G is connected, k .G/ D n  kC1 2 2 we can consider G as a graph obtained by deleting some edges from the complete graph Kn . If G D Kn n M such that jMj  kC1 , then it follows from Lemma 5.1.3 2 kC1 . u t that k .G/ < n  2 , a contradiction. So G D Kn n M, where 0  jMj  k1 2 Remark 5.1.7 ([90]). The graphs with k .G/ D n  dk=2e or k .G/ D n  dk=2e have been characterized by Theorems 5.1.5 and 5.1.6. A natural question is, for the lower bounds, whether we can characterize the graphs with k .G/ D 1 or k .G/ D 1. It seems not easy to solve such a problem. Note that the minimal graphs with k .G/ D 1 or k .G/ D 1 are the trees of order n. So, an interesting problem could be what is the maximal graphs with k .G/ D 1 or k .G/ D 1? Actually, one can check that a connected graph G obtained from the complete graph Kn1 by attaching a pendant edge is a such graph, which is obviously a unique maximum such graph. However, to characterize all the maximal graphs remains unsolved. Here maximal (minimal) means that adding (deleting) any edge will destroy k .G/ D 1 or k .G/ D 1, whereas maximum means a such graph that has the largest number of edges.

5.2 Graphs with the Second Largest Generalized Connectivity As a continuation of their investigation, Li and Mao later turned their attention to characterizing graphs with k .G/ D n  dk=2e  1 and k .G/ D n  dk=2e  1. One may notice that k .G/ D ndk=2e if and only if G itself is the complete graph Kn for k even. So for k even, it is possible to continue to characterize k .G/ D ndk=2e1 by deleting more edges from the complete graph Kn . Theorem 5.2.1 ([91]). Let n and k be two integers such that k is even and 4  k  n and G be a connected graph of order n. Then k .G/ D n  2k  1 if and only if G D Kn n M where M is an edge set such that 1  .Kn ŒM/  2k and 1  jMj  k  1. Different from the proof of Theorem 5.1.5, in order to find n  2k  1 internally N they designed a procedure to emphasize disjoint S-Steiner trees in GŒS [ GŒS; S, seeking for some edges “evenly” in GŒS and let them together with the edges of N form n  k internally disjoint Steiner trees T1 ; T2 ;    ; Tnk with its root EG ŒS; S

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N respectively. Applying this procedure designed by them, w1 ; w2 ;    ; wnk 2 S, Li and Mao [91] proved that the remaining edges in GŒS can form k2 spanning 2 k2 trees, which are also 2 internally disjoint S-Steiner trees. These trees together with T1 ; T2 ;    ; Tnk are n  2k  1 internally disjoint S-Steiner trees, accomplishing the proof of the above theorem. By Theorem 5.2.1 and Observation 2.1, they obtained the following theorem for k .G/. Theorem 5.2.2 ([91]). Let n and k be two integers such that k is even and 2  k  n and G be a connected graph of order n. Then k .G/ D n  2k  1 if and only if G D Kn n M where M is an edge set satisfying one of the following conditions: (1) .Kn ŒM/ D 1 and 1  jMj  b n2 c; (2) 2  .Kn ŒM/  2k and 1  jMj  k  1. By the Nash-Williams-Tutte theorem, they luckily characterized the graphs with k .G/ D n  dk=2e  1 and k .G/ D n  dk=2e  1 for k even. But, for k odd, it is not easy to characterize the graphs with k .G/ D n  dk=2e  1. So, Li, Li, Mao, and Sun [75] considered the case k D 3 and obtained the following result. Theorem 5.2.3 ([75]). Let G be a connected graph of order n .n  3/. Then 3 .G/ D n3 if and only if G is a graph satisfying one of the following conditions: • • • •

G D P4 [ .n  4/K1 ; G D P3 [ rP2 [ .n  2r  3/K1 .r D 0; 1/; G D C3 [ rP2 [ .n  2r  3/K1 .r D 0; 1/; G D sP2 [ .n  2s/K1 .2  s  b n2 c/. But, for the edge case, Li and Mao [89] showed that the statement is different.

Theorem 5.2.4 ([89]). Let G be a connected graph of order n .n  3/. Then 3 .G/ D n  3 if and only if G is a graph satisfying one of the following conditions: • • • •

G D rP2 [ .n  2r/K1 .2  r  b n2 c/; G D P4 [ sP2 [ .n  2s  4/K1 .0  s  b n4 c/; 2 n3 G D P3 [ tP2 [ .n  2t  3/K1 .0  t  b 2 c/; G D C3 [ tP2 [ .n  2t  3/K1 .0  t  b n3 c/. 2

5.3 The Minimal Size of a Graph with Given Generalized k-Connectivity Li and Li and Shi [82] determined the minimal number of edges among graphs with 3 .G/ D 2, i.e., graphs G of order n and size e.G/ with 3 .G/ D 2.

5.3 The Minimal Size of a Graph with Given Generalized k-Connectivity

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Theorem 5.3.1 ([82]). If G is a graph of order n with 3 .G/ D 2, then e.G/  d 65 ne. Moreover, the lower bound is sharp for all n  4 but n D 9; 10, whereas the best lower bound for n D 9; 10 is d 65 ne C 1. Proof. Since 3 .G/ D 2, it follows from Proposition 4.1.10 that ı.G/  2 and any two vertices of degree 2 are not adjacent. Denote by X the set of vertices of degree 2. We have that X is a stable set. Put Y D V.G/ n X and obviously there are 2jXj edges joining X to Y. Assume that m0 is the number of edges joining two vertices belonging to Y. It is clear that e D 2jXj C m0 :

(5.1)

Since every vertex of Y has degree at least 3 in G, it follows that 2jXj C 2m0  3jYj D 3.n  jXj/, and hence

P v2Y

d.v/ D

5jXj C 2m0  3n:

(5.2)

Combining (5.1) with (5.2), we have 5 5 5 e D .2jXj C m0 / D 5jXj C m0  5jXj C 2m0  3n; 2 2 2 and hence e  65 n. Since the number of edges is an integer, it follows that e  d 65 ne. t u They constructed a graph class to show that the bound of Theorem 5.3.1 is sharp. Example 5.3. For a positive integer t ¤ 2, let C D x1 y1 x2 y2    x2t y2t x1 be a cycle of length 4t. Add t new vertices z1 ; z2 ;    ; zt to C, and join zi to xi and xiCt , for 1  i  t. The resulting graph is denoted by H. Then 3 .H/ D 2; see Fig. 5.1. Li and Mao [89] considered a generalization of the above problem. Let s.n; k; `/ and t.n; k; `/ denote the minimal number of edges of a graph G of order n with k .G/ D ` .1  `  n  dk=2e/ and k .G/ D ` .1  `  n  dk=2e/, respectively. From Theorem 5.3.1, one can see that s.n; 3; 2/ D d 65 ne for all n  4 but n D 9; 10. From Theorems 5.1.5 and 5.1.6, we know that ( n for k even s.n; k; n  dk=2e/ D t.n; k; n  dk=2e/ D 2n k1  2 for k odd 2 Fig. 5.1 The graph H with 3 .H/ D 2

y2t

x2t

yt+3 xt+3 yt+2

x1 y1 x2 y2

zt

xt+2 yt+1

z1 z2 x3

y3

z3

xt yt

xt+1

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From Theorems 5.2.1 and 5.2.2, we know that for k even ! n s.n; k; n  dk=2e  1/ D kC1 2 and ! jnk n t.n; k; n  dk=2e  1/ D :  2 2 Li and Mao [89] investigated t.n; 3; `/ and derived the following result. Theorem 5.3.2 ([89]). Let n be an integer with n  3. Then:  (1) t.n; 3; n  2/ D n2  1; n c; (2) t.n; 3; n  3/ D 2  b nC3 2 (3) t.n; 3; 1/ D n˙  1;  (4) t.n; 3; `/  `.`C1/ n for n  11 and 2  `  n  4. Moreover, the bound is 2`C1 sharp. The complete bipartite graph G D K`;`C1 is a sharp example for the bound of Theorem 5.3.2. Recall that an `-connected graph G is called minimally `-connected if the graph G n e is not `-connected for any edge e of G. For the generalized k-connectivity, similarly, a graph G is called minimally .k; `/-generalized connected if the generalized k-connectivity of G is `, but the generalized k-connectivity of G n e is less than ` for any edge e of G. Though it is easy to find the sharp lower bound of e.G/, very little progress has been made on the sharp upper bound. S. Li first posed a problem on minimally .3; 2/-generalized connected graphs and showed that 2n  4  e.G/  3n  10 if G is a minimally .3; 2/-generalized connected graph. We now pose the following general problem. Open Problem. Let G be a graph of order n and size e.G/ such that G is minimally .k; `/-generalized connected. Find a sharp upper bound of e.G/.

Chapter 6

Nordhaus-Gaddum-Type Results

Let G .n/ denote the class of simple graphs of order n and G .n; m/ the subclass of G .n/ having graphs with n vertices and m edges. Given a graph parameter f .G/ and a positive integer n, the Nordhaus-Gaddum Problem is to determine sharp bounds for .1/ f .G/ C f .G/ and .2/ f .G/  f .G/, as G ranges over the class G .n/, and characterize the extremal graphs. The Nordhaus-Gaddum-type relations have received wide attention; see a survey paper [3] by Aouchiche and Hansen. Alavi and Mitchem in [2] and Hellwig and Volkmann in [60] investigated Nordhaus-Gaddum-type results for the classical connectivity and edge-connectivity in G .n/. Achuthan and Achuthan [1] considered the same problem in G .n; m/. Nordhaus-Gaddum inequalities for edge toughness were given in 1995 by Peng, Chen, and Koh [118].

6.1 Results for Graphs in G .n/ Li and Mao [86] investigated the Nordhaus-Gaddum-type relations on the generalized edge-connectivity. They first focused on the graphs in G .n/. Theorem 6.1.1 ([86]). Let G 2 G .n/ and k; n be two integers with 2  k  n. Then: (1) 1  k .G/ C k .G/  n  dk=2e;

2 (2) 0  k .G/  k .G/  ndk=2e . 2 Moreover, the upper and lower bounds are sharp. The following observation indicates the graphs attaining the above lower bound. Observation 6.1 ([86]). Let k; n be two integers with 2  k  n. Then k .G/  k .G/ D 0 if and only if G or G is disconnected.

© The Author(s) 2016 X. Li, Y. Mao, Generalized Connectivity of Graphs, SpringerBriefs in Mathematics, DOI 10.1007/978-3-319-33828-6_6

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6 Nordhaus-Gaddum-Type Results

a

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A4n

Fig. 6.1 The graph classes Ani .1  i  4/

For n  5, An1 is a graph class as shown in Fig. 6.1a, each graph G of which satisfies that .G/ D 1 and dG .v1 / D n  1, where v1 2 V.G/; An2 is a graph class as shown in Fig. 6.1b; each graph G of which satisfies that .G/ D 2 and dG .u1 / D n  1, where u1 2 V.G/; An3 is a graph class as shown in Fig. 6.1c; each graph G of which satisfies that .G/ D 2 and dG .v1 / D n  1, where v1 2 V.G/; An4 is a graph class as shown in Fig. 6.1d; and each graph G of which satisfies that .G/ D 2. As we know, it is not easy to characterize the graphs with k .G/ D 1, even with 3 .G/ D 1. So Li and Mao wanted to add some conditions to attack such a problem. Motivated by such an idea, they hope to characterize the graphs with k .G/ C k .G/ D 1. Actually, the Nordhaus-Gaddum problems also need to characterize the extremal graphs attaining the bounds. The following observation and lemma are preparations for Proposition 6.1.3. C For n  5, let K2;n2 be the graph obtained from the complete bipartite graph CC denote K2;n2 by adding one edge on the part having n  2 vertices, and let K2;n2 any of the two graphs which are obtained from K2;n2 by adding two edges on the part having n  2 vertices. Observation 6.2 ([86]). Let n be an integer with n  5. Then: CC (1) n .K2;n2 /  2; C C /  2, n .K2;n2 / D 1; (2) n1 .K2;n2 (3) n2 .K2;n2 /  2, n .K2;n2 / D n1 .K2;n2 / D 1. CC Proof. (1) As shown in Fig. 6.2a, we have n .K2;n2 /  2. C C /j D 2.n  (2) As shown in Fig. 6.2b, we have n1 .K2;n2 /  2. Since jE.K2;n2 2.n2/C1 C C C 2/C1 and n .K2;n2 /  b n1 c, it follows that n .K2;n2 /  1. Since K2;n2 C is connected, it follows that n .K2;n2 / D 1. (3) As shown in Fig. 6.2c, we have n2 .K2;n2 /  2. Let U D fu1 ; u2 g and W D fw1 ; w2 ;    ; wn2 g be the two parts of the complete bipartite graph K2;n2 . Choose S D fu1 ; u2 ; w1 ; w2 ;    ; wn3 g. If there exists an S-Steiner tree containing the vertex wn2 , then this tree will use n  1 edges of E.K2;n2 /, and hence n1 .K2;n2 /  1 since jE.K2;n2 /j D 2.n  2/. Suppose that any S-Steiner tree does not contain the vertex wn2 . Pick up such a tree, say T. Then there exists a vertex with degree 2 in T, which implies that there is no other

6.1 Results for Graphs in G .n/

a

u1

wi

w1

69

u2

wi+1 wj

u1

wj+1 wn−2

w1

u2

wi−1 wi

w2

wi+1 wn−2

++ K2,n−2

b w1

u1

wi wi+1 wn−3 wn−2 w1 w2

w2

c

u1

w1

u1

u2

w2

u2

wn−3 wn−2

u1

u2

wi+1 wn−3 wn−2 w1

wi

w2

u2

wi wi+1 wn−3 wn−2

+ K2,n−2

u1

w1 wi−1 wi

u2

wi+1 wn−2

u1

u2

w1 wi−1 wi wi+1 wj−1wj wj+1wn−2

K2,n−2

Fig. 6.2 Graphs for Observation 6.2

S-Steiner tree in K2;n2 . So n1 .K2;n2 /  1. Since K2;n2 is connected, it follows that n1 .K2;n2 / D 1. From Proposition 4.2.7, n .K2;n2 / D 1. t u Lemma 6.1.2 ([86]). Let G be a connected graph of order n, and let k be an integer with 3  k  n. If .G/ D 3 and there exists a vertex u 2 V.G/ such that dG .u/ D n  1, then k .G/  2 for 3  k  n. Proof. Let G1 ;    ; Gr be the connected components of G  u. Since .G/ D 3, it follows that ı.Gi /  2 .1  i  r/. Let jV.Gi /j D ni .1  i  r/ and V.Gi / D fvi;1 ; vi;2 ;    ; vi;ni g. Then there exists an edge, without loss of generality, say ei D vi;1 vi;2 2 E.Gi / such that Gi n ei is connected for 1  i  r. Thus Gi n ei contains a spanning tree, say Ti .1  i  r/. The tree T induced by the edges in fuv1;1 ; uv2;1 ;    ; uvr;1 g [ E.T1 / [ E.T2 / [    [ E.Tr / and the tree T 0 induced by the edges in fv1;1 v1;2 ; uv1;2 ;    ; uv1;n1 g [ fv2;1 v2;2 ; uv2;2 ;    ; uv2;n2 g [    [ fvr;1 vr;2 ; uvr;2 ;    ; uvr;nr g are two spanning trees of G, and hence n .G/  2. Combining this with Proposition 4.2.7, we get k .G/  2 for 3  k  n. t u Proposition 6.1.3 ([86]). Let G be a graph of order n, and let k be an integer with 3  k  n. Then k .G/ C k .G/ D 1 if and only if G (symmetrically, G) satisfies one of the following conditions: (1) G 2 An1 or G 2 An2 ; (2) G 2 An3 and there exists a component Gi of G  v1 such that Gi is a tree and jV.Gi /j < k; C ; K2;n2 g for k D n and n  5, G 2 fP3 ; C3 g for k D n D 3, (3) G 2 fK2;n2 G 2 fC4 ; K4 n eg for k D n D 4, G D K3;3 for k D n D 6, G D K2;n2 for k D n  1 and n  5, or G D C4 for k D n  1 D 3. Proof. Let G be a graph satisfying one of the conditions .1/, .2/, and .3/. One can see that G is connected and its complement G is disconnected. Thus k .G/ C

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6 Nordhaus-Gaddum-Type Results

k .G/ D k .G/ and k .G/  1. We only need to show that k .G/  1 for each graph G satisfying one of the conditions .1/, .2/, and .3/. For G 2 An1 , since ı.G/ D 1, we have k .G/  1 by Observation 2.1. For G 2 An2 , it follows from Proposition 4.2.8 that k .G/  ı.G/  1 D 1 since dG .v1 / D dG .v2 / D ı.G/ D 2. Suppose that G 2 An3 and there exists a connected component Gi of Gv1 such that Gi is a tree and jV.Gi /j < k. Set V.Gi / D fvi;1 ; vi;2 ;    ; vi;ni g. We choose S  V.G/ such that V.Gi / [ fv1 g D S0  S. Then jE.GŒS0 /j D 2ni  1. Since every spanning tree of GŒS0  uses ni edges of E.GŒS0 /, there exists at most one spanning tree in GŒS0 , which implies that there is at most one Steiner tree connecting S in G. So k .G/  1. C For G D K2;n2 , we have n .G/ D 1 by .2/ of Observation 6.2. If G D K2;n2 , then it follows from .3/ of Observation 6.2 that n .K2;n2 / D n1 .K2;n2 / D 1. For c D b 95 c D 1. For G 2 fP3 ; C3 ; C4 ; K4 n eg, one can G D K3;3 , n .G/  b jE.G/j n1 check that k .G/  1 for k D n or k D n  1. From these together with k .G/  1, we have k .G/ C k .G/ D k .G/ D 1. Conversely, we suppose k .G/ C k .G/ D 1. Then k .G/ D 1 and k .G/ D 0, or k .G/ D 1 and k .G/ D 0. By symmetry, without loss of generality, let k .G/ D 1 and k .G/ D 0. From these together with Proposition 4.2.5, .G/ D 0 and 1  .G/  3. So we have the following three cases to consider: Case 1. .G/ D 1. For n D 3, one can check that G D P3 satisfies .G/ D 1 but .G/ D 0. Now we assume n  4. Since .G/ D 1, there exists a cut edge in G, say e D u1 v1 . Let G1 and G2 be the two connected components of G n e such that u1 2 V.G1 / and v1 2 V.G2 /. Set V.G1 / D fu1 ; u2 ;    ; un1 g and V.G2 / D fv1 ; v2 ;    ; vn2 g, where n1 C n2 D n. Suppose ni  2 .i D 1; 2/. For any ui ; uj 2 V.G1 /, ui and uj are connected in G since there exists a path ui v2 uj in G; for any vi ; vj 2 V.G2 /, vi and vj are connected in G since there exists a path vi u2 vj in G; for any ui 2 V.G1 / and vj 2 V.G2 / (i ¤ 1 or j ¤ 1), vi vj 2 E.G/. Clearly, the path u1 v2 u2 v1 connects u1 and v1 in G. So G is connected, a contradiction. Thus n1 D 1 or n2 D 1. Without loss of generality, let n1 D 1. Then V.G1 / D fu1 g and V.G2 / D fv1 ; v2 ;    ; vn1 g. Clearly, G is a graph obtained from G2 by attaching the edge e D u1 v1 . Since u1 vj … E.G/ .1 < j  n  1/, it follows that u1 vj 2 E.G/. If dG .v1 /  n  2, then there exists a vertex vj such that v1 vj 2 E.G/, which results in .G/  1, a contradiction. So dG .v1 / D n  1 and G 2 An1 ; see Fig. 6.1a. Case 2. .G/ D 2. For n D 3; 4, the graph G 2 fC3 ; C4 ; K4 n eg satisfies that .G/ D 2 and .G/ D 0. Since 3 .C3 / D 1, 3 .C4 / D 1, 4 .C4 / D 1, 3 .K4 n e/ D 2, and 4 .K4 n e/ D 1, we have G D C3 for k D n D 3, G 2 fC4 ; K4 n eg for k D n D 4, and G D C4 for k D n1 D 3. Now we assume n  5. Since .G/ D 2, there exists an edge cut M such that jMj D 2. Let G1 and G2 be the two connected components of G n M, V.G1 / D fu1 ;    ; un1 g and V.G2 / D fv1 ;    ; vn2 g, where n1 C n2 D n. Clearly, GŒM D 2K2 or GŒM D P3 . At first, we consider the case GŒM D 2K2 . Without loss of generality, let M D fu1 v1 ; u2 v2 g. Since n  5, it follows that n1  3 or n2  3. Without loss

6.1 Results for Graphs in G .n/

71

of generality, let n1  3. Clearly, any two vertices vi ; vj 2 V.G2 / are connected in G since there exists a path vi u3 vj in G. Furthermore, for any ui 2 V.G1 /, ui v1 2 E.G/ or ui v2 2 E.G/. So G is connected and .G/  1, a contradiction. Next, we consider the case GŒM D P3 . Without loss of generality, let P D v1 u1 v2 be the path of order 3. Since n  5, there exist at least two vertices in G  fu1 ; v1 ; v2 g. If n1  2 and n2  3, then we can check that G is connected, a contradiction. So we assume n1 D 1 or n2 D 2, that is, V.G2 / D fv1 ; v2 g or V.G1 / D fu1 g. For the former, V.G1 / D fu1 ; u2 ;    ; un2 g. Since .G/ D 2, it follows that v1 v2 2 E.G/. Clearly, v1 uj ; v2 uj … E.G/ .2  j  n  2/, which implies that v1 uj ; v2 uj 2 E.G/. Therefore, u1 uj … E.G/ .2  j  n  2/ since G is disconnected. Thus u1 uj 2 E.G/ for each j .2  j  n  2/, and hence dG .u1 / D n  1 and G 2 An2 ; see Fig. 6.1b. For the latter, let V.G2 / D fv1 ; v2 ;    ; vn1 g. We first consider the case v1 v2 2 E.G/. Since u1 vj … E.G/ .3  j  n  1/, we have u1 vj 2 E.G/. If 3  dG .v1 /  n  2 and 3  dG .v2 /  n  2, then there exist two vertices vi and vj such that v1 vi ; v2 vj 2 E.G/ .3  i; j  n  1/, and hence G is connected, a contradiction. So dG .v1 / D n  1 or dG .v2 / D n  1. Without loss of generality, let dG .v1 / D n  1. Thus G 2 An3 ; see Fig. 6.1c. Now we focus on the graph G  v1 . Let G1 ; G2 ;    ; Gr be the connected components of G  v1 and V.Gi / D fvi;1 ; vi;2 ;    ; vi;ni g .1  i  r/, where Pr n D n  1. If there exists some connected component Gi such that Gi D K2 , iD1 i then G 2 An2 ; see Fig. 6.1b. So we assume ni  3. Then we show the following claim and get a contradiction. Claim 1. For each connected component Gi of G  v1 , if ni  k or ni  k  1 and jE.Gi /j  ni , then k .G/  2 for 3  k  n. Proof of Claim 1. For an arbitrary S  V.G/ with jSj D k, we only prove that .S/  2 for v1 … S. The case v1 2 S can be proved similarly. If there exists some connected component Gi such that S D V.Gi /, then ni D k, and Gi has a spanning tree, say Ti . It is also a Steiner tree connecting S. Since the tree Ti0 induced by the edges in fv1 vi;1 ; v1 vi;2 ;    ; v1 vi;ni g is another S-Steiner tree and Ti ; Ti0 are two edge-disjoint S-Steiner trees, it follows that .S/  2. Assume now S ¤ V.Gi / for ni  k .1  i  r/. S PrLet Si D S \ V.Gi / .1  i  r/ and jSi j D ki . It is clear that r S D S and iD1 i iD1 ki D k. Thus Si  V.Gi / for each connected component Gi such that ni  k, and Sj  V.Gj / for each connected component Gj such that nj  k  1 and jE.Gj /j  nj . We will show that there are two edge-disjoint Steiner trees connecting Si [ fv1 g in GŒSi [ fv1 g for each i .1  i  r/ so that we can combine these trees to form two edge-disjoint Steiner trees connecting S in G. Suppose that Gi is a connected component such that ni  k. Note that V.Gi / D fvi;1 ; vi;2 ;    ; vi;ni g. Since Si  V.Gi /, there exists a vertex, without loss of generality, say vi;1 , such that 0 vi;1 … Si . Clearly, Gi contains a spanning tree, say Ti;1 . Thus the tree Ti;1 induced 0 by the edges in fv1 vi;1 g [ E.Ti;1 / is a Steiner tree connecting Si [ fv1 g in GŒGi [ fv1 g. Since the tree Ti;2 induced by the edges in fv1 vi;2 ; v1 vi;3 ;    ; v1 vi;ni g is another

72

6 Nordhaus-Gaddum-Type Results

Steiner tree connecting Si [ fv1 g, clearly, Ti;1 and Ti;2 are edge disjoint. Assume that Gj is a connected component such that nj  k  1 and jE.Gj /j  nj . Note that V.Gj / D fvj;1 ; vj;2 ;    ; vj;nj g. Then there exists an edge, without loss of generality, say ej D vj;1 vj;2 2 E.Gj / such that Gj n ej contains a spanning tree of Gj , say 0 0 Tj;1 . Thus the tree Tj;1 induced by the edges in fv1 vj;1 g [ E.Tj;1 / and the tree Tj;2 induced by the edges in fvj;1 vj;2 ; v1 vj;2 ;    ; v1 vj;nj g are two edge-disjoint Steiner trees connecting Sj [ fv1 g. Now we combine these small Steiner trees connecting Si [ fv1 g .1  i  r/ by the vertex v1 to form two big Steiner trees connecting S. It is clear that the tree T1 induced by the edges in E.T1;1 / [ E.T2;1 / [    [ E.Tr;1 / and the tree T2 induced by the edges in E.T1;2 / [ E.T2;2 / [    [ E.Tr;2 / are our desired trees, and hence .S/  2. From the arbitrariness of S, we have k .G/  2. t u By Claim 1, we know that G 2 An3 and there exists a connected component Gi of G  v1 such that ni  k  1 and Gi is a tree. We next consider the case v1 v2 … E.G/; see Fig. 6.1d. Thus v1 v2 2 E.G/. Since u1 vj … E.G/ .3  j  n  1/, it follows that u1 vj 2 E.G/, and hence v1 vj ; v2 vj … E.G/ since G is disconnected. Thus v1 vj ; v2 vj 2 E.G/ for each j .3  j  n1/. Let CC R D fvj j 3  j  n  1g. If jE.GŒR/j  2, then G contains a subgraph K2;n2 , and hence n .G/  2 by .1/ of Observation 6.2. Combining this with Proposition 4.2.7, k .G/  2 for 3  k  n, a contradiction. If jE.GŒR/j < 2, then G D K2;n2 C C and K2;n2 . From Observation 6.2 and Proposition 4.2.7, we have k .K2;n2 /2 for 3  k  n  1 and k .K2;n2 /  2 for 3  k  n  2, a contradiction. So C C G D K2;n2 for k D n, G D K2;n2 for k D n, or G D K2;n2 for k D n  1. Case 3. .G/ D 3. For n D 4, we have G D K4 , and hence 3 .G/ D 4 .G/ D 2 by Theorem 2.1.2, that is, k .G/  2, a contradiction. Assume n  5. Since .G/ D 3, there exists an edge cut M such that jMj D 3. Let G1 and G2 be the two connected components of GnM, V.G1 / D fu1 ; u2 ;    ; un1 g and V.G2 / D fv1 ; v2 ;    ; vn2 g, where n1 Cn2 D n. Clearly, GŒM D P4 or GŒM D P3 [ K2 or GŒM D 3K2 or GŒM D K1;3 . For the former three cases, ni  3 .i D 1; 2/ and n  6 since .G/ D 3. To shorten the discussion, we only show .G/  1 for GŒM D P4 and get a contradiction among the former three cases. Without loss of generality, let GŒM D P4 D u1 v1 u2 v2 . For any ui ; uj 2 V.G1 / .1  i  n1 /, ui and uj are connected in G since there exists a path ui v3 uj in G; for any vi ; vj 2 V.G2 / .1  i  n2 /, vi and vj are connected in G since there exists a path vi u3 vj in G; for any ui 2 V.G1 / and vj 2 V.G2 / (i ¤ 3 and j ¤ 3), ui and vj are connected in G since there exists a path ui v3 u3 vj in G. Since u3 vj 2 E.G/ .1  j  n2 / and v3 ui 2 E.G/ .1  i  n1 /, it follows that G is connected, as desired. Now we consider the graph G such that GŒM D K1;3 . Assume n1  2. If n2  4, then we can check that G is connected and get a contradiction. Therefore, n2 D 3, V.G2 / D fv1 ; v2 ; v3 g, and V.G1 / D fu1 ; u2    ; un3 g. Since .G/ D 3, it follows that v1 v2 ; v2 v3 ; v1 v3 2 E.G/. Since vi uj … E.G/ .1  i  3; 2  j  n3/, we have vi uj 2 E.G/. If there exists some vertex uj .2  j  n  3/ such that u1 uj 2 E.G/,

6.1 Results for Graphs in G .n/

73

Fig. 6.3 Graphs for case 3 of Proposition 6.1.4

a G1

G2 v1

u2

u3

u1

un−3

d

G2 v1

G1 u1

Fig. 6.4 Graphs for n D 6 in case 3 of Proposition 6.1.4

v2 v3

a

v2

G1 u1

v3

e v4 v5

G1 u1

vn−1

b

A1

b

c

G2 v1

v4 v5

v2 v3

u1

v2 v3

vn−1

f

G2 v1

v4 v5

v2 v3

c

A2

G2 v1

G1

A3

G2 v1

G1 u1

vn−1

v2 v3

d

e

v4 v5 vn−1

v4 v5 vn−1

A4

then G is connected, a contradiction. So u1 uj 2 E.G/ for 2  j  n  3. Thus dG .u1 / D n  1; see Fig. 6.3a. From Lemma 6.1.2, k .G/  2 for 3  k  n since .G/ D 3, a contradiction. We now assume n1 D 1. Then V.G1 / D fu1 g and V.G2 / D fv1 ; v2    ; vn1 g. If GŒfv1 ; v2 ; v3 g D 3K1 or GŒfv1 ; v2 ; v3 g D K1 [ K2 , then we have u1 vj 2 E.G/ since u1 vj … E.G/ .4  j  n  1/. From this together with the fact that G is disconnected and v1 v3 ; v2 v3 2 E.G/ and vi vj … E.G/ .1  i  3; 4  j  n  1/, we have vi vj 2 E.G/ .1  i  3; 4  j  n  1/. Thus G contains a complete bipartite graph K3;n3 as its subgraph; see Fig. 6.3b,c. From Table 2.1, n .G/ D b 3.n3/ c  2 for n1 n  7, which implies that k .G/  2 for 3  k  n and n  7. Since .G/ D 3, it follows that n  6, and hence we only need to consider the case n D 6. Thus G D Ai .1  i  4/; see Fig. 6.4. If G D Ai .2  i  4/, then n .G/  2 for k D n D 6; see Fig. 6.4b,c,d. Therefore k .G/  2 for 3  k  6. If G D A1 , then n .G/  b jE.G/j c D b 95 c D 1 for k D n D 6. For k D 5, we can check that n1 3 .G/  4 .G/  5 .G/  2; see Fig. 6.4e. So G D K3;3 for k D n D 6. Suppose GŒfv1 ; v2 ; v3 g D P3 . Without loss of generality, let v1 v2 ; v2 v3 2 E.G/. If 3  dG .v2 /  n2 (see Fig. 6.3d), then there exists at least one vertex vj such that v2 vj 2 E.G/, and hence v1 vj ; v3 vj … E.G/ .4  j  n  1/ since u1 vj 2 E.G/ .4  j  n  1/, v1 v3 2 E.G/, and G is disconnected. Thus, v1 vt ; v3 vt 2 E.G/ for each t .4  t  n  1/. Since d.v4 /  ı.G/  .G/ D 3, we have v4 v2 2 E.G/, or there exists some vertex vj .5  j  n  1/ such that v4 vj 2 E.G/, which implies that CC G contains a subgraph K2;n2 and so n .G/  2 by .1/ of Observation 6.2. From Proposition 4.2.7, k .G/  2 for 3  k  n, a contradiction. If dG .v2 / D n  1 (see Fig. 6.3e), then it follows from Lemma 6.1.2 that k .G/  2 for 3  k  n, since .G/ D 3, a contradiction. Assume that GŒfv1 ; v2 ; v3 g D K3 . Without loss of generality, let v1 v2 ; v1 v3 ; v2 v3 2 E.G/. If dG .v1 / D n  1 or dG .v2 / D n  1 or dG .v3 / D n  1 (see Fig. 6.3f),

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6 Nordhaus-Gaddum-Type Results

then it follows from Lemma 6.1.2 that k .G/  2 for 3  k  n since .G/ D 3, a contradiction. If 3  dG .vi /  n  2 .1  i  3/, then G is connected, another contradiction. t u If one of G and G is disconnected, we can characterize the graphs attaining the upper bound by Theorem 5.1.6. Proposition 6.1.4 ([86]). Let G be a graph of order n, and let k be an integer with 3  k  n. If G is disconnected, then k .G/ C k .G/ D n  dk=2e if and only if G D Kn for k even; G D Kn n M for k odd, where M is an edge set such that 0  jMj  k1 . 2 If both G and G are connected, we can obtain a structural property of the graphs attaining the upper bound although it seems too difficult to characterize them. Proposition 6.1.5 ([86]). Let G be a graph of order n, and let k be an integer with 3  k  n. If k .G/ C k .G/ D n  dk=2e, then .G/  ı.G/  dk=2e  1. One can see that the graphs with k .G/Ck .G/ D ndk=2e must have a uniform degree distribution. By this property, Li and Mao constructed a graph class to show that the two upper bounds of Theorem 6.1.1 are sharp for k D n. Example 6.1. Let n; r be two positive integers such that n D 4rC1. From Table 2.1, we know that n .K2r;2rC1 / D n .K2r;2rC1 / D r. Let E be the set of the edges of these r spanning trees in K2r;2rC1 . Then there remain 2r.2r C 1/  4r2 D 2r edges in K2r;2rC1 except for the edges in E . Let M be the set of these 2r edges. Set G D K2r;2rC1 n M. Then n .G/ D r, M  E.G/, and G is a graph obtained from two cliques K2r and K2rC1 by adding 2r edges in M between them, that is, one endvertex of each edge belongs to K2r and the other endvertex belongs to K2rC1 . Note that E.G/ D E.K2r / [ M [ E.K2rC1 /. Now we show that n .G/  r. As we know, K2r contains r Hamilton paths, say P1 ; P2 ;    ; Pr , and so does K2rC1 , say P01 ; P02 ;    ; P0r . Pick up r edges from M, say e1 ; e2 ;    ; er . Let Ti be the tree induced by the edges in E.Pi / [ E.P0i / [ fei g .1  i  r/. Then T1 ; T2 ;    ; Tr are r spanning trees in G,     namely, n .G/  r. Since jE.G/j D 2r2 C 2rC1 C2r D 4r2 C2r and each spanning 2 2 tree uses 4r edges, these edges can form at most b 4r 4rC2r c D r spanning trees, that D n  d n2 e and is, n .G/  r. So n .G/ D r. Clearly, n .G/ C n .G/ D 2r D n1 2 ndn=2e 2 2 n .G/  n .G/ D r D . 2 Li, Mao, and Sun [90] were concerned with analogous inequalities involving the generalized k-connectivity for the graphs in G .n/. Theorem 6.1.8 ([90]). Let G 2 G .n/ and k; n be two integers with 2  k  n. Then: (1) 1  k .G/ C k .G/  n  dk=2e; (2) 0  k .G/  k .G/  Œ ndk=2e 2 . 2 Moreover, the upper and lower bounds are sharp.

6.2 Results for Graphs in G .n; m/

75

Proof. (1) To avoid confusion, we denote the generalized local connectivity of a N D Kn , for any k-subset S, k-subset S in a graph G by .GI S/. Since G [ G we have .GI S/ C .GI S/  .Kn I S/. Suppose that k .Kn / D .Kn I S0 / for some k-subset S0 . Then we have k .Kn / D .Kn I S0 /  .GI S0 / C .GI S0 /  k .G/Ck .G/. This together with k .Kn / D ndk=2e results in k .G/Ck .G/  n  dk=2e. If k .G/ C k .G/ D 0, then k .G/ D k .G/ D 0. Thus G and G are all disconnected, which is impossible. So k .G/ C k .G/  1. (2) It follows immediately from .1/. Since n .G/ D n .G/ for any graph G of order n, we have the following remark for generalized (edge-)connectivity. Remark 6.1.9 ([90]). The above example only shows that the upper bounds of Theorems 6.1.1 and 6.1.8 are sharp for the case k D n. A natural question is to find examples showing that the upper bounds of Theorems 6.1.1 and 6.1.8 are sharp for each k with 3  k < n. Note that the complete graph G D Kn can attain the upper bound of .1/, but clearly G is disconnected. Therefore, when we require that both G and G are connected, is there a graph which can attain the upper bounds of Theorem 6.1.1 and 6.1.8, respectively, or simultaneously for each k with 3  k  n?

6.2 Results for Graphs in G .n; m/ Li and Mao also focused on the graphs in G .n; m/ in [86]. Let us begin with another problem, called the maximum connectivity. It was pointed out by Harary [58] that given the number of vertices and edges of a graph, the largest connectivity possible can also be read out of the inequality .G/  .G/  ı.G/.  Theorem 6.2.1 ([58]). For each n; m with 0  n  1  m  n2 , .G/  .G/ 

j 2m k ; n

where the maximum is taken over all graphs G 2 G .n; m/. Li and Mao considered the similar problem for the generalized edge-connectivity. Corollary 6.2.2 ([86]). For any graph G 2 G .n; m/ and 3  k  n, k .G/ D 0 for m < n  1; k .G/  b 2m c for m  n  1. n Although the above bound of k .G/ is the same as .G/, the graphs attaining the upper bound seem to be very rare. Actually, we can obtain some structural properties of these graphs. Proposition 6.2.3 ([86]). For any G 2 G .n; m/ and 3  k  n, if k .G/ D b 2m c n for m  n  1, then: •

2m n

is not an integer;

76

6 Nordhaus-Gaddum-Type Results

• ı.G/ D b 2m c; n • For u; v 2 V.G/ such that dG .u/ D dG .v/ D b 2m c, uv … E.G/. n The following corollary is immediate. Corollary 6.2.4. For any graph G with n vertices and m edges, if then k .G/  2m  1. n

2m n

is an integer,

Achuthan, Achuthan, and Caccetta [1] restricted their attention to the subclass of G .n; m/ consisting of graphs with n vertices and m edges. They investigated the edge-connectivity, diameter, and chromatic number parameters. For the edgeconnectivity .G/, they showed that .G/ C .G/  maxf1; n  1  mg. In this section, we consider a similar problem on the generalized edge-connectivity. By Theorem 1.4.1 and Corollary 6.2.2, Li and Mao derived the following theorem. Theorem 6.2.5 ([86]). Let G 2 G .n; m/, and let k be an integer with 3  k  n. For n  6, we have: (1) L.n; m/  k .G/ C k .G/  M.n; m/, (2) 0  k .G/  k .G/  N.n; m/, where ( L.n; m/ D

if b n3 c C 1  m  maxf1; b 12 .n  2  m/cg 2m cg if 0  m  b n3 c minfn  2m  1; b n2  n1

n 2

8 ˆ n  d 2k e ˆ ˆ ˆ ˆ
k, there exist k C 1 internally disjoint paths Pi .1  i  k C 1/ connecting x to y0 in G.v1 / by Menger’s theorem. Note that at most one of them is a path of length 1. Let PkC1 be such a path if xy0 2 E.G.v1 //, and let k0 ; k1 ; : : : ; k` be integers such that 0 D k0 < k1 <   
k, x has k neighbors, say x1 ; x2 ; : : : ; xk , in G.v1 /. Let Pi be the path xxi , and let k0 ; k1 ; : : : ; k` be integers such that 0 D k0 < k1 <    < k` D k. From Proposition 7.1.1, we can construct S i .ki  ki1 C 1/ internally disjoint S-Steiner trees Ti;ji .1  ji  ki  ki1 C 1/ in . kjDk Pj /Ti for each i, where 1  i  `. Note i1 C1 that Ti;ji and Tr;jr are internally disjoint for i ¤ r. Thus Ti;ji .1  i  `; 1  ji  ki  ki1 C 1/ are .k C `/ internally disjoint S-Steiner trees. t u They also showed that the bounds of .1/ and .2/ in Theorem 7.1.4 are sharp. Let Kn be a complete graph with n vertices and Pm be a path with m vertices, where m  2. Since 3 .Pm / D 1 and 3 .Kn / D n  2, it is easy to see that 3 .Kn Pm / D n  1. Thus, Kn Pm is a sharp example for .2/. For .1/, Example 7.1 is a sharp one. Li and Mao [88] derived the following upper bound of 3 .GH/ from Theorems 4.1.1, 4.1.7, and 7.2. Theorem 7.1.5 ([88]). Let G and H be two connected graphs. Then 3 .GH/  minfb 43 3 .G/ C r1  43 d r21 ecjV.H/j; b 43 3 .H/ C r2  43 d r22 ecjV.G/j; ı.G/ C ı.H/g, where r1 .G/ .mod 4/ and r2 .H/ .mod 4/. Moreover, the bound is sharp. For the generalized 3-edge-connectivity, Sun [127] obtained the following result. Theorem 7.1.6 ([127]). Let G and H be two connected graphs, we have 3 .GH/  3 .G/ C 3 .H/. Moreover, the bound is sharp. Using Fan lemma [136, page 170] and expansion lemma [136, page 162], Li and Mao [88] obtained the following lower bound of 3 .G ı H/, which could be seen as an extension of Yang and Xu’s result. Theorem 7.1.7 ([88]). Let G and H be two connected graphs. Then 3 .G ı H/  3 .G/jV.H/j: Moreover, the bound is sharp. For a tree T and a connected graph H, they showed that 3 .T ı H/ D jV.H/j, which can be seen as an improvement of Theorem 7.1.7. From Theorem 7.1.4, one may wonder whether 3 .TH/ D 3 .T/ C 3 .H/  1 for a connected graph H and a tree T (note that 3 .T/ D .T/ D 1). For example, let T D P3 and H D K4 . Then 3 .T/ D .T/ D 1 and 3 .H/ D 2. One can check that 3 .TH/ D 3 > 2 D 3 .T/ C 3 .H/  1. So the equality does not hold for the Cartesian product of a tree and a connected graph. In [129], Sun determined the generalized 3-(edge)connectivity for Cartesian products of various combinations of complete graphs, cycles, wheels, and trees.

7.2 Results for the Spanning-Tree Packing Number

83

For the lexicographic product, Li and Mao [88] derived an upper bound of 3 .G ı H/ from Propositions 4.2.4 and 4.1.7 and Theorem 7.3. Theorem 7.1.8 ([88]). Let G and H be two connected graphs. If G is nontrivial and noncomplete, then 3 .G ı H/  b 43 3 .G/ C r  43 d 2r ecjV.H/j, where r

.G/ .mod 4/. Moreover, the bound is sharp. The graph Pn ı P3 .n  4/ indicates that both the lower bound of Theorem 7.1.7 and the upper bound of Theorem 7.1.8 are sharp. For 3 .G ı H/, Li, Yue, and Zhao in [96] obtained the following lower and upper bounds. Theorem 7.1.9 ([96]). Let G and H be a connected graph. Then 3 .G ı H/  3 .H/ C 3 .G/jV.H/j: Moreover, the lower bound is sharp. Theorem 7.1.10 ([96]). Let G be a connected graph and H be a nontrivial graph. Then 3 .G ı H/  minfb 43 .G/C2 cjV.H/j2 ; ı.H/ C ı.G/jV.H/jg. Moreover, the 3 upper bound is sharp. The graph Pt ı Pnt is a sharp example for both Theorem 7.1.9 and Theorem 7.1.10. For general k, to get similar results for product graphs is difficult. One may try to think about the case for k D 4 or 5.

7.2 Results for the Spanning-Tree Packing Number In [119], Peng and Tay determined the spanning-tree packing numbers of Cartesian products of various combinations of complete graphs, cycles, and complete multipartite graphs; see Table 2.1 for more details. Ku, Wang, and Hung [67] studied the problem of constructing the maximum number of edge-disjoint spanning trees in Cartesian product networks. Let G D .VG ; EG / be a graph having `1 edge-disjoint spanning trees and H D .VH ; EH / be a graph having `2 edge-disjoint spanning trees. Two methods were proposed for constructing edge-disjoint spanning trees in GH. The graph G has r1 D jEG j`1 .jVG j1/ more edges than that are necessary for constructing `1 edgedisjoint spanning trees in it, and the graph H has r2 D jEH j`2 .jVH j1/ more edges than that are necessary for constructing `2 edge-disjoint spanning trees in it. By assuming that r1  `1 and r2  `2 , their first method shows that `1 C`2 edge-disjoint spanning trees can be constructed in GH. Their second construction does not need any assumption on G and H. It constructs `1 C `2  1 edge-disjoint spanning trees in GH by efficiently combining the edge-disjoint spanning trees of G and H. From the aspect of the number of constructed edge-disjoint spanning trees, the first construction is optimal for the case r1 D `1 and r2 D `2 . And the second is optimal for

84

7 Results for Graph Products

the case r1  `1 and r2 D 0. The usefulness of the constructions was demonstrated by applying them to determine the maximum numbers of edge-disjoint spanning trees in many important product networks, including tori, torus-connected k-ary hypercubes, hypercubes, mesh-connected k-ary hypercubes, meshes, generalized hypercubes, mesh-connected trees, and hyper Petersen networks. For the graph H D .VH ; EH /, let T be a set of ` edge-disjoint spanning trees in H. There are jEH j  `.jVH j  1/ edges in H that are not contained by any spanning tree of T . Those edges are nontree edges of H with respect to T . A different set of ` edge-disjoint spanning trees defines a different set of nontree edges. However, for any set of ` edge-disjoint spanning trees, there are always jEH j  `.jVH j  1/ nontree edges. To simplify the description, we say that a graph H has ` edge-disjoint spanning trees and r nontree edges if H has ` edge-disjoint spanning trees and jEH j D `.jVH j  1/ C r. Also for easy description, for any subset U  VH , the subgraph .U; ¿/ is simply denoted by U; and for any vertex v 2 VH , the subgraph .fvg; ¿/ is simply denoted by v. Let G D .VG ; EG / be a graph having `1  1 edge-disjoint spanning trees and r1  0 nontree edges, and let H D .VH ; EH / be a graph having `2  1 edge-disjoint spanning trees and r2  0 nontree edges. We denote X D fX1 ; X2 ; : : : ; X`1 g as a set of `1 edge-disjoint spanning trees in G and Y D fY1 ; Y2 ; : : : ; Y`2 g as a set of `2 edge-disjoint spanning trees in H. The subgraph of G whose vertex set is VG and edge set is the set of nontree edges with respect to X is denoted by NG . Similarly, the subgraph of H whose vertex set is VH and edge set is the set of nontree edges with respect to Y is denoted by NH . For convenience, the above notation is summarized in Table 7.1. In the remainder of this section, some preliminary results are presented. Lemma 7.2.1 ([67]). Let F D .V; E/ be a graph with jVj  2. If F has ` edgedisjoint spanning trees, then jVj  2`. Lemma 7.2.2 ([67]). Let F D .V; E/ be a graph with jVj  2. If F has ` edgedisjoint spanning trees and t > 0 nontree edges, then jVj  2` C 1. Table 7.1 A summary of notations Notation G D .VG ; EG / X D fX1 ; X2 ; : : : ; X`1 g NG H D .VH ; EH / Y D fY1 ; Y2 ; : : : ; Y`2 g NH

Description A graph having `1  1 edge-disjoint spanning trees and r1  0 nontree edges A set of `1 edge-disjoint spanning trees in G The subgraph of G whose vertex set is VG and edge set is the set of r1 nontree edges with respect to X A graph having `2  1 edge-disjoint spanning trees and t2  0 nontree edges A set of `2 edge-disjoint spanning trees in H The subgraph of H whose vertex set is VH and edge set is the set of r2 nontree edges with respect to Y

7.2 Results for the Spanning-Tree Packing Number

85

Lemma 7.2.3 ([67]). Let F D .V; E/ be a graph having t > 0 edges. If F contains no cycle, then there exists a subset U  V of t vertices such that every vertex in U has a path to a vertex in V  U. Proof. Since F contains no cycle, all connected components in it are trees. By combining this and F having t edges, it is easy to see that the number of connected components in F is jVj  t. Let C1 ; C2 ; : : : ; CjVjt be the components. From every Ci .1  i  jVj  t/, we arbitrarily select a vertex vi . We set U D V  fv1 ; v2 ; : : : ; vjVjt g. Clearly, jUj D t. Consider a vertex u 2 U. Let Cs be the component containing u, where 1  s  jVjt. Since Cs is a connected component, the vertex u has a path to the vertex vs , which is in V  U D fv1 ; v2 ; : : : ; vjVjt g. Thus, every vertex in U has a path to a vertex in V  U. t u We assume that r1 D `1 and r2 D `2 show that `1 C `2 edge-disjoint spanning trees can be constructed in GH. The proposed construction can be applied easily to the cases r1  `1 and r2  `2 by simply ignoring some nontree edges. Lemma 7.2.4 ([67]). Let F D .V; E/ be a graph having ` edge-disjoint spanning trees and ` nontree edges. Then there exists a set T of ` edge-disjoint spanning trees in F such that the set of nontree edges (with respect to T ) alone does not constitute any cycle in F. Proof. All cycles mentioned in this proof are simple ones. Let T D fT1 ; T2 ; : : : ; T` g be an arbitrary set of ` edge-disjoint spanning trees in F. Let N.T / be the subgraph whose vertex set is V and edge set is the set of ` nontree edges with respect to T . If N.T / contains no cycle, the lemma holds. In the following, we assume that there are cycles in N.T / and prove this lemma by transforming T into another set of ` edge-disjoint spanning trees to satisfy the condition. The idea of the transformation is to let some spanning tree in T exchange an edge with N.T / to get a new set T 0 of ` edge-disjoint spanning trees such that N.T 0 / has less cycles. Then, we set T D T 0 and repeat the transformation until N.T / contains no cycle. The new set T 0 is obtained as follows: Let C be a cycle in N.T / and uv be an edge in the cycle. N.T / may contain several connected components. Let K D .VK ; EK / be the connected component that includes C. Since K is a connected graph having at most ` edges and there is a cycle in K, we can easily conclude that jVK j  ` and, thus, jVK  fu; vgj  `  2. Since uv is a nontree edge, it is not contained by any spanning tree in T . Therefore, for each Ti .1  i  `/, if we add the edge uv to it, a cycle Ci D : : : wi uv : : : occurs. Note that wi … fu; vg, 1  i  `. Since all spanning trees Ti .1  i  `/ are edge disjoint, all wi are different from each other; otherwise there are two Ti containing the same edge wi u. Since jVK  fu; vgj  `  2, wi … fu; vg and every wi is distinct, 1  i  `, there exists a wj … VK , 1  j  `. We transform T into T 0 D .T  fTj g/ [ fTj0 g, where Tj0 is obtained from Tj by exchanging wj u for uv. Clearly, Tj0 is still a spanning tree, and Tj0 does not share any edge with all the other Ti , 1  i  ` and i ¤ j. Thus, T 0 is a set of ` edge-disjoint spanning trees. The graph N.T 0 / is obtained from N.T / by removing uv and adding wj u. The removal of uv reduces at least one cycle from N.T /. Since wj … VK , it follows that wj is contained by a connected component

86

7 Results for Graph Products

K 0 ¤ K of N.T /. Thus, the effect of adding wj u into N.T / is just to connect K and K 0 together. Clearly, the connection creates no cycle. Therefore, N.T 0 / contains less cycles than N.T /, which completes the proof. t u According to Lemma 7.2.4, we may assume without loss of any generality that both NG and NH contain no cycle. Consider the subgraph NG . Its vertex set is VG . Since NG has r1 D `1 edges and contains no cycle, according to Lemma 7.2.3, there is a subset of VG , denoted by UG , such that jUG j D `1 and in the subgraph NG , every vertex in UG has a path to a vertex in VG  UG . Similarly, there is a subset of VH , denoted by UH , such that jUH j D `2 and in the subgraph NH , every vertex in UH has a path to a vertex in VH  UH . We note the properties of UG , NG , UH , and NH as follows: Property 7.2.5 ([67]). jUG j D `1 , jUH j D `2 , every vertex in UG is connected to a vertex in VG  UG via only those nontree edges in NG , and every vertex in UH is connected to a vertex in VH  UH via only those nontree edges in NH . Now, we are ready to describe the construction of `1 C `2 edge-disjoint spanning trees in GH. The first `1 1 spanning trees are obtained by using fX2 ; X3 ; : : : ; X`1 g and Y1 as follows: Let t be an arbitrary vertex in UG . We compose a set of `1  1 pairs .i; u/, where 2  i  `1 and u 2 UG  ftg, such that no two pairs have the same i or u. Then, for each composed pair .i; u/, a spanning tree .Xi VH / [ .uY1 / is constructed; see Fig. 7.2a. The next `2  1 spanning trees are obtained similarly by using fY2 ; Y3 ; : : : ; Y`2 g and X1 . Let t0 be an arbitrary vertex in UH . We compose a set of `2  1 pairs of .i; u/, where 2  i  `2 and u 2 UH  ft0 g, such that no two pairs have the same i or u. Then, for each composed pair .i; u/, a spanning tree .VG Yi / [ .X1 u/ is constructed; see Fig. 7.2b. The next spanning tree is constructed by using X1 , Y1 , and NH , which is

a

b

VH − UH

UH

UH u=

u=t

VH − UH t Yi

UG

Yi

Y1

Yi

VG − UG

Yi

Xi

Xi

Xi

Xi

X1

Yi

Fig. 7.2 (a) The first `1  1 spanning trees, 2  i  `1 and u 2 UG  ftg. (b) The first `2  1 spanning trees, 2  i  `2 and u 2 UH  ft0 g

7.2 Results for the Spanning-Tree Packing Number

a

b

VH − UH

UH

87

UH

VH − UH

t

t

connected by edges in NG

Y1

UG

VG − UG Y1

connected by edges in NH

X1

X1

X1

Y1

Fig. 7.3 (a) The next spanning tree. (b) The last spanning tree

.X1 .VH  UH // [ .tY1 / [ ..VG  ftg/NH /I see Fig. 7.3a. The last spanning tree is constructed by using X1 , Y1 , and NG , which is ..VG  UG /Y1 / [ .X1 t0 / [ .NG .VH  ft0 g//I see Fig. 7.3b. Theorem 7.2.6 ([67]). Let G be a graph having `1 edge-disjoint spanning trees and r1 nontree edges and H be a graph having `2 edge-disjoint spanning trees and r2 nontree edges. If r1 D `1 and r2 D `2 , the product GH has `1 C `2 edge-disjoint spanning trees and `1 C `2 nontree edges. Proof. It is not difficult to check that the `1 C `2 spanning trees constructed above are mutually edge disjoint. There are jVG j  `1 and jVH j  `2 edges, respectively, in G and H. Thus, there are jVG j  jVH j  .`1 C `2 / edges in GH. Each of the constructed trees has .jVG j  jVH j  1/ edges. Thus, there are jVG j  jVH j  .`1 C `2 /  .jVG j  jVH j  1/.`1 C `2 / D `1 C `2 nontree edges. t u Proposition 7.2.7 ([67]). The number of constructed edge-disjoint spanning trees in Theorem 7.2.6 is the maximum. Proof. By Theorem 7.2.6, the number of nontree edges of GH is `1 C`2 after a set of `1 C `2 edge-disjoint spanning trees are constructed. According to Lemma 7.2.1, we have jVG jjVH j  2.`1 C `2 /, from which it is easy to conclude that `1 C `2 edges are not sufficient to construct a spanning tree in GH. t u It was assumed in the first construction that G and H, respectively, have r1 D `1 and r2 D `2 nontree edges. In the following, we show that `1 C `2  1 edge-disjoint spanning trees can be constructed in GH without using any nontree edges of G and H. Theorem 7.2.8 ([67]). Let G be a graph having `1 edge-disjoint spanning trees and H be a graph having `2 edge-disjoint spanning trees. Then the product network GH has `1 C `2  1 edge-disjoint spanning trees.

88

7 Results for Graph Products

Corollary 7.2.9 ([67]). Let G be a graph having `1 edge-disjoint spanning trees and no nontree edge. Let H be a graph having `2 edge-disjoint spanning trees and no nontree edge. Then the product network GH has `1 C `2  1 edge-disjoint spanning trees and .jVG j  `1 /.jVH j  `2 /  .`1  1/.`2  1/ nontree edges. The number `1 C `2  1 is the maximum. Corollary 7.2.10 ([67]). Let G be a graph having `1 edge-disjoint spanning trees and H be a graph having `2 edge-disjoint spanning trees. Then the product network GH has `1 C`2 1 edge-disjoint spanning trees and jVG jjVH j1C`1 C`2 `2 jVG j nontree edges. The number `1 C `2  1 is the maximum. Like the work of [67] for Cartesian product, Li, Li, Mao, and Yue [76] focused their attention on the case k D n and investigated the spanning-tree packing number of lexicographic product graphs by graph decomposition and a result on Hamilton decomposition from [68]. Theorem 7.2.11 ([76]). Let G and H be two connected graphs, and let n .G/ D k, n .H/ D `, jV.G/j D n1 .n1  2/, and jV.H/j D n2 .n2  2/. Then: (1) If kn2 D `n1 , then n .G ı H/  kn2 .Dl`n1 /; m (2) If `n1 > kn2 , then n .G ı H/  kn2  knn211 C `  1; m l 2 2 (3) If `n1 < kn2 , then n .G ı H/  kn2  2 kn C `. n1 C1 Moreover, the lower bounds are sharp. To show the sharpness of the above lower bounds of Theorem 7.2.11, they considered the following example. Example 7.2. (1) Let G and H be two connected graphs with jV.G/j D n1 and jV.H/j D n2 which can be decomposed into exactly k and ` edge-disjoint spanning trees of G and H, respectively, satisfying kn2 D `n1 . From .1/ of Theorem 7.2.11, n .G ı H/  kn2 D `n1 . Since jE.G ı H/j D jE.H/jn1 C jE.G/jn22 D `.n2 1/n1 Ck.n1 1/n22 D kn2 .n2 1/Ck.n1 1/n22 D kn2 .n1 n2  1/, we have n .G ı H/  jE.GıH/j D kn2 . Then n .G ı H/ D kn2 D `n1 . So the n1 n2 1 lower bound of .1/ is sharp. (2) Consider the graphs G D P3 and H D K4 . Clearly, n .G/ D k D 1, n .H/ D ` D 2, jV.G/j D n1 D 3, jV.H/j D n2 D 4, jE.G/j D 2, jE.H/j D 6, and 6 D `n1 > kn2 D 4. On the one hand, we have n .G ı H/  kn2  d knn211 e C e D 4 by .2/ of Theorem 7.2.11. On the other hand, `  1 D 4  1 C 2  d 41 3 D b 50 c D 4. So n .GıH/ D 4. jE.GıH/j D 50 and hence n .GıH/  jE.GıH/j n1 n2 1 11 So the lower bound of .2/ is sharp. (3) Consider the graphs G D P2 and H D P3 . Clearly, n .G/ D k D 1, n .H/ D ` D 1, jV.G/j D n1 D 2, jV.H/j D n2 D 3, jE.G/j D 1, jE.H/j D 2, and 2 2 D `n1 < kn2 D 3. On the one hand, n .GıH/  kn2 2d nkn eC` D 2 by .3/ 1 C1 of Theorem 7.2.11. On the other hand, jE.GıH/j D jE.H/jn1 CjE.G/jn22 D 13. Then n .G ı H/  jE.GıH/j D b 13 c D 2. So n .G ı H/ D 2 and the lower bound n1 n2 1 5 of .3/ is sharp.

Chapter 8

Maximum Generalized Local Connectivity

In this chapter, we introduce the results on the extremal problems of the generalized connectivity and generalized edge-connectivity. Recall that .G/ D minfG .x; y/ j x; y 2 V.G/; x ¤ yg is the connectivity of G. In contrast to this parameter, .G/ D maxfG .x; y/ j x; y 2 V.G/; x ¤ yg, introduced by Bollobás, is called the maximum local connectivity of G. The problem of determining the smallest number of edges, h.nI   r/, which guarantees that any graph with n vertices and h.nI   r/ edges will contain a pair of vertices joined by r internally disjoint paths was posed by Erdös and Gallai; see [7] for details. Bollobás [14] considered the problem of determining the largest number of edges, f .nI   `/, for graphs with n vertices and local connectivity at most `, that is, f .nI   `/ D maxfe.G/ j jV.G/j D n and .G/  `g. One can see that h1 .nI   ` C 1/ D f .nI   `/ C 1. Similarly, let G .x; y/ denote the local edge-connectivity between x and y in G. Then .G/ D minfG .x; y/ j x; y 2 V.G/; x ¤ yg and .G/ D maxfG .x; y/ j x; y 2 V.G/; x ¤ yg are the edge-connectivity and maximum local edge-connectivity, respectively. So the edge version of the above problems can be given similarly. Set g.nI   `/ D maxfe.G/ j jV.G/j D n and .G/  `g. Let h2 .nI   r/ denote the smallest number of edges which guarantees that any graph with n vertices and h2 .nI   r/ edges will contain a pair of vertices joined by r edge-disjoint paths. Similarly, h2 .nI   ` C 1/ D g.nI   `/ C 1. The problem of determining the precise value of the parameters f .nI   `/, g.nI   `/, h1 .nI   r/, or h2 .nI   r/ has obtained wide attention, and many results have been worked out; see [14–16, 70–72, 101, 102, 132]. Similar to the classical maximum local connectivity, Li and Li and Mao [74] introduced the parameter  k .G/ D maxf.S/ j S  V.G/; jSj D kg, which is called the maximum generalized local connectivity of G. There they considered the problem of determining the largest number of edges, f .nI  k  `/, for graphs with n vertices and maximal generalized local connectivity at most `, that is, f .nI  k  `/ D maxfe.G/ j jV.G/j D n and  k .G/  `g. They also considered the smallest number of edges, h1 .nI  k  r/, which guarantees that any graph with n vertices and © The Author(s) 2016 X. Li, Y. Mao, Generalized Connectivity of Graphs, SpringerBriefs in Mathematics, DOI 10.1007/978-3-319-33828-6_8

89

90

8 Maximum Generalized Local Connectivity

h1 .nI  k  r/ edges will contain a set S of k vertices such that there are r internally disjoint S-Steiner trees. It is easy to check that h1 .nI  k  ` C 1/ D f .nI  k  `/ C 1 for 0  `  n  dk=2e  1. The edge version of these problems are also introduced and investigated by Li and Mao in [87]. Similarly, g.nI k  `/ D maxfe.G/ j jV.G/j D n and k .G/  `g, and h2 .nI k  r/ is the smallest number of edges, h2 .nI k  r/, which guarantees that any graph with n vertices and h2 .nI k  r/ edges will contain a set S of k vertices such that there are r edge-disjoint S-Steiner trees, and also similarly, h2 .nI k  ` C 1/ D g.nI k  `/ C 1 for 0  `  n  dk=2e  1.

8.1 Results for the Maximum Generalized Local Connectivity In order to make the parameter f .nI k  `/ meaningful, we need to determine the range of `. In fact, with the help of the definitions of  k .G/, k .G/, k .G/, and k .G/ and Theorems 2.1.1 and 2.1.2, Li and Mao made the following observation, which implies that 1  `  n  dk=2e. Observation 8.1 ([87]). Let k; n be two integers with 3  k  n. Then for a connected graph G of order n, 1   k .G/  k .G/  n  dk=2e. Moreover, the upper and lower bounds are sharp. Note that n .G/ D n .G/ D  n .G/ D n .G/. From Theorems 5.1.5 and 5.1.6, we can derive the following corollary. Corollary 8.1.1 ([87]). For a connected graph G of order n, n .G/ D  n .G/ D n .G/ D n .G/ D bn=2c if and only if G D Kn for n even or G D Kn n M for n odd, where M is a set of edges such that 0  jMj  n1 . 2 A minor of a graph G is any graph obtainable from G by means of a sequence of vertex and edge deletions and edge contractions. For connected regular graphs, Sun [128] got the following results. Proposition 8.1.2 ([128]). For a connected `-regular graph G,  k .G/ D ` if and only if Kk;` is a minor of G. Proof. We assume that .S/ D  k .G/ where S  V.G/ and jSj D k. Let T D fTj j 1  j  `g be a set of ` internally disjoint S-Steiner trees. Since G is `-regular, it follows that every vertex of S is a leaf in each Tj , and hence Kk;1 is a minor of each S Tj for 1  j  `. Thus, Kk;` is a minor of the subgraph `jD1 Tj of G. Let H D Kk;` be a minor of G. Then H can be formed from G by deleting edges and vertices and by contracting edges such that V.H/ D A [ B where A D fui j 1  i  kg  V.G/ is one part of the bipartition of V.H/ and B is another one. Then, with an inverse procedure, we can obtain the original graph G from H and get ` internally disjoint Steiner trees connecting A. Thus, .A/  ` and we have  k .G/ D .A/ D ` since .A/  `. t u

8.1 Results for the Maximum Generalized Local Connectivity

91

For a cubic connected graph G, equality holds in Observation 8.1. Proposition 8.1.3 ([128]). For a cubic connected graph G, we have  k .G/ D k .G/: Proof. Since the result clearly holds for the case that k .G/ D 1, we assume that k .G/  2 in the following argument. Suppose that .S/ D k .G/ D ` where S  V.G/ and jSj D k. Let T D fTj j 1  j  `g be a set of ` edge-disjoint S-Steiner trees. For any two trees Tj1 and Tj2 where 1  j1 ¤ j2  `, if they have a common vertex, say v, which does not belong to S, then we must have dG .v/  4, a contradiction. Thus, any two Steiner trees in T are internally disjoint, so  k .G/  .S/  ` D .S/ D k .G/. Together with Observation 8.1, we complete the proof. t u The Cartesian product P.2h/ D Ch P2 of a cycle Ch of length h  3 and P2 is called a prism. Suppose that V.P.2h// D f.ui ; vj / W 1  i  h; j D 1; 2g and E.P.2h// D f.ui ; v1 /.ui ; v2 / W 1  i  hg [ f.ui ; v1 /.uiC1 ; v1 / W 1  i  hg [ f.uj ; v2 /.ujC1 ; v2 / W 1  j  hg where the subscripts are taken by modulo h. Then, the Möbius ladder M.2h/ of order 2h is the graph obtained from P.2h/ by deleting the edges .u1 ; v1 /.vh ; v1 / and .u1 ; v2 /.vh ; v2 / and adding the edges .u1 ; v1 /.vh ; v2 / and .u1 ; v2 /.vh ; v1 /. Sun in [128] obtained the precise values of  3 .G/ and 3 .G/ if G is a cubic connected Cayley graph on an Abelian group. Theorem 8.1.4 ([128]). If G is a cubic connected Cayley graph on an Abelian group, then   3 .G/ D 3 .G/ D

3 if G is a Möbius ladder 2 otherwise:

Based on this result and the monotonicity of  k .G/ and k .G/, Sun derived the following result. Theorem 8.1.5 ([128]). Let G be a cubic connected Cayley graph on an Abelian group of order n  9: (i) If G is a prism, then there exists an integer k1 such that  k .G/ D k .G/ D 2 for any 3  k  k1 and  k .G/ D k .G/ D 1 for any k1 C 1  k  n; moreover, we have n2 C 2  k1  d 3nC2 e. 4 (ii) If G is a Möbius ladder, then there exist two integers k2 ; k3 such that  k .G/ D k .G/ D 3 for any 3  k  k2 ,  k .G/ D k .G/ D 2 for any k2 C 1  k  k3 , and  k .G/ D k .G/ D 1 for any k3 C 1  k  n; moreover, we have 3  k2  d 3n e and n2 C 2  k3  d 3nC2 e. 8 4

92

8 Maximum Generalized Local Connectivity

8.2 Graphs with at Most Two Internally Disjoint Steiner Trees Connecting Any Three Vertices Let us now introduce a graph class Gn by a few steps. Li, Li, and Mao [74] first introduced the following graph operation. Let H be a connected graph and u be a vertex of H. They defined the attaching operation at the vertex u on H as follows: (i) Identifying u and a vertex of a K4 ; (ii) u is attached with only one K4 . The vertex u is called an attaching vertex. For r  3, Gr D fHri j 1  i  7g is a class of graphs of order r; see Fig. 8.1 for details. Let Hni .1  i  7/ be the class of graphs, each of them is obtained from a graph i Hr by the attaching operation at some vertices of degree 2 on Hri , where 3  r  n and 1  i  7 (note that Hni 2 Hni ). Furthermore, Gn is another class of graphs that S contains Gn , given as follows: G3 D fK3 g, G4 D fK4 g, G5 D fB1 g [ . 7iD1 H5i /, S S S G6 D fB3 ; B4 g [ . 7iD1 H6i /, G7 D 7iD1 H7i , G8 D fB2 g [ . 7iD1 H8i /, and S Gn D 7iD1 Hni for n  9; see Fig. 8.2 for details. Li and Li and Mao obtained the following theorem: Theorem 8.2.1 ([74]). Let G be a connected graph of order n such that  3 .G/  2. Then  2n  2 if n D 4 e.G/  2n  3 if n  3; n ¤ 4 with equality if and only if G 2 Gn . Fig. 8.1 The graph class Gr Hr1

Hr2

Hr3

Hr4

Hr5

Hr6

Hr7

K4

Fig. 8.2 The graphs for Theorem 8.2.1 B1

B2

B3

B4

8.2 Graphs with at Most Two Internally Disjoint Steiner Trees Connecting Any. . . Fig. 8.3 Graphs obtained from H51 and H53

a

b

d

c

B7

B6

B5

e

B8

93

f

B9

B10

We now give the proof of Theorem 8.2.1. The following three observations are immediate. Observation 8.2 ([74]). Let G and H be two connected graphs and H 0 be a subdivision of H. If H 0 is a subgraph of G and  3 .H/  3, then  3 .G/  3. Observation 8.3 ([74]). Let H be a graph, u and v be two vertices in H, and G be a graph obtained from H by attaching a K4 at u. If there are three internally disjoint paths between u and v in H, then  3 .G/  3. Observation 8.4 ([74]). For each graph in Fig. 8.3,  3  3. Lemma 8.2.2 ([74]). Let G be a graph containing a clique K4 as its subgraph. If there exists a path connecting two vertices of K4 in G n E.GŒK4 /, then  3 .G/  3. Lemma 8.2.3 ([74]). Let G be a graph obtained from H54 by adding a vertex x and two edges xy; xz, where y; z 2 V.H54 /. Then  3 .G/  3 or G D H65 . Lemma 8.2.4 ([74]). For any connected graph G with order 5 and size 8,  3 .G/  3. Lemma 8.2.5 ([74]). For any connected graph G of order 5 and size 7,  3 .G/  2 and G 2 fB1 ; H51 ; H53 ; H54 g. Lemma 8.2.6 ([74]). For any connected graph G with order 6 and size 10,  3 .G/  3. Lemma 8.2.7 ([74]). Let G be a connected graph of order 6 and size 9. If  3 .G/  2, then G 2 fB3 ; B4 g or G 2 fH61 ; H62 ; H65 g or G 2 H63 . In Lemma 8.2.4 through Lemma 8.2.7, we have dealt with the cases for n  6. Now we assume n  7. Lemma 8.2.8 ([74]). Let G0 be a graph obtained from G by deleting a vertex of  degree 2. If G0 2 Gn1 (n  7), then G 2 Gn or  3 .G/  3. Proof. Let x be the deleted vertex of degree 2 in G. Since n  7, it follows that G0 … fK3 ; K4 ; B1 g. From Observation 8.3 and Lemma 8.2.2, if G0 2 fB2 ; B3 ; B4 g, then we can check that G 2 H93 or  3 .G/  3. From now on, we consider G0 2  Gn1  fB2 ; B3 ; B4 g:

94

8 Maximum Generalized Local Connectivity 1 Case 1. G0 2 Hn1 .

1 Suppose that there is no K4 in G0 . Then G0 D Hn1 . Since n  7, it follows that 1 1 G D Hn 2 Hn or G must contain a B7 or B10 as its subgraph, and hence G 2 Gn or  3 .G/  3 by Observation 8.2. Suppose that there exists at least one K4 in G0 . For each K4 , if NG .x/\V.K4 y/ ¤ ¿, then it follows from Lemma 8.2.2 that  3 .G/  3, where y is an attaching vertex in G0 . Suppose that NG .x/ \ V.K4  y/ D ¿ for all K4  G0 . Clearly, we can 1 consider the graph G0 2 Hn1 as the joint of K2 and r isolated vertices and then doing the attaching operation at some vertices of degree 2 on K2 _ rK1 . So we consider NG .x/  K2 _ rK1 .r  1/. For r  3, we have G 2 Hn1 , or G contains the graph B7 or B10 as its subgraph, which implies that G 2 Gn or  3 .G/  3. For r D 2, from Lemma 8.2.2, we only need to consider NG .x/  V.K2 _ 2K1 /. By Observation 8.3, G 2 H111 or G 2 H81 or G 2 H83 or  3 .G/  3. For r D 1, K2 _ K1 is a triangle, and G0 is a graph obtained from this triangle by the attaching operation at some vertices of this triangle since n  7. From Observation 8.3 and Lemma 8.2.2, we have  3 .G/  3. 2 3 Case 2. G0 2 Hn1 or G0 2 Hn1 . 2 We only prove the conclusion for G0 2 Hn1 , since the same can be showed 3 2 0 for G 2 Hn1 analogously. Without loss of generality, let Hn1 be the graph class 2 obtained from Hr by the attaching operation at some vertices of degree 2 on Hr2 , where r D n  1; n  4; n  7. One can see that u1 and vr=2 can be the attaching vertices. From Lemma 8.2.2, we only need to consider the case that NG .x/  V.Hr2 /. Set NG .x/ D fx1 ; x2 g. Thus x1 ; x2 2 V.Hr2 /. If dHr2 .x1 / D dHr2 .x2 / D 2, without loss of generality, let x1 D u1 and x2 D vr=2 ; then neither u1 nor vr=2 is an attaching vertex by Observation 8.3. We can choose a path P WD u3 u4    ur=2 vr=2 xu1 connecting u1 and u3 in G  fu2 ; v1 ; v2 g. Thus, G contains a subdivision of B6 as its subgraph (see Figs. 8.3 and 8.4a), and hence  3 .G/  3. If dHr2 .x1 / D 2 and dHr2 .x2 / D 3, without loss of generality, let x1 D u1 ; then we can find a path connecting u1 and u3 and obtain  3 .G/  3 for x2 2 V.Hr /  fu2 ; v1 ; v2 g. For x2 D u2 and x2 D v2 , G contains a B5 and B8 as its subgraph, and hence  3 .G/  3. If x2 D v1 , then G 2 Hn3 and so G 2 Gn . For 3  dHr2 .xi /  4 .i D 1; 2/, one can check that G contains a subdivision of one of fB5 ; B6 ; B8 ; B9 ; B10 g, and hence  3 .G/  3.

Fig. 8.4 Graphs for Lemma 8.2.8

x

a

P

u1

u2

u3

ur/2

v1

v2

v3

vr/2

b w1

w2

w3

u1

u2

ur/2

v1

v2

vr/2

8.2 Graphs with at Most Two Internally Disjoint Steiner Trees Connecting Any. . .

95

4 5 Case 3. G0 2 Hn1 or G0 2 Hn1 .

Note that only vr=2 can be an attaching vertex in Hr4 (see Fig. 8.4b), where r D n  1; n  4. From Lemma 8.2.2, we only need to consider NG .x/  V.Hr4 /. We can 2 by identifying one edge u1 v1 in consider Hr4 as a graph obtained from H54 and Hr3 each of them. Consider NG .x/ \ fw1 ; w2 ; w3 g ¤ ¿. If NG .x/ ¤ fw2 ; v1 g, then G contains a subdivision of one element in fBi j 11  i  16g as its subgraph, where each Bi is a graph obtained from H54 by adding a vertex x and two edges xy; xz such that y; z 2 V.H54 /. From Lemma 8.2.3, we have  3 .G/  3. If NG .x/ D fw2 ; v1 g, then one can also get that  3 .G/  3. Now we can assume that NG .x/\fw1 ; w2 ; w3 g D ¿. For jfu1 ; v1 g \ NG .x/j D 2, G contains a B10 as its subgraph, and hence  3 .G/  3. For jfur=2 ; vr=2 g \ NG .x/j D 2, if vr=2 is not an attaching vertex in Hr4 , then G 2 Hn5 ; if vr=2 is an attaching vertex in Hr4 , then it follows from Observation 8.3 that  3 .G/  3. For the other cases, we can also check that  3 .G/  3. 6 7 Case 4. G0 2 Hn1 or G0 2 Hn1 .

From the above case 2 and Lemma 8.2.3, we can get  3 .G/  3 in this case.

t u

Similarly, we have the following lemma. Lemma 8.2.9 ([74]). Let G0 be a graph obtained from G by deleting a vertex of  degree 3. If G0 2 Gn1 (n  7), then  3 .G/  3. Lemma 8.2.10 ([74]). Let G be a graph obtained from G0 by deleting an edge e D x1 x2 and adding a vertex x such that NG .x/ D fx1 ; x2 ; x3 g, where x3 2 .V.G0 /   fx1 ; x2 g/. If G0 2 Gn1 (n  7), then G 2 Gn or  3 .G/  3. Proof. Since n  7, it follows that G0 … fK3 ; K4 ; B1 g. From Observation 8.3 and Lemma 8.2.2, if G0 2 fB2 ; B3 ; B4 g, then we can easily check that  3 .G/  3 or  G 2 Gn . Thus we consider G0 2 .Gn1  fB2 ; B3 ; B4 g/. We claim that if there exists a K4 in G0 such that e 2 E.K4 /, then  3 .G/  3. Let V.K4 / D fu1 ; u2 ; u3 ; u4 g. Without loss of generality, let x1 D u2 and x2 D u4 . If x3 2 V.K4 /, then x3 D u1 or x3 D u3 , and hence  3 .G/  3; see Fig. 8.5a. So we assume that x3 … V.K4 /. From Lemma 8.2.2, if x3 belongs to another clique of order 4 such that x3 is not an attaching vertex, then  3 .G/  3, or G 2 Gn . So we only need to consider x3 2 V.Hri / .1  i  7/. If neither u2 nor u4 is an attaching vertex, then u1 or u3 is an attaching vertex, say u1 . Therefore, there must exist a path P connecting x3 and u1 such that u2 ; u3 ; u4 … V.P/ since Hri .1  i  7/ is connected. Then the tree T1 induced by the edges in fxu2 ; xu4 g [ E.P/, the tree T2 induced by Fig. 8.5 Graphs for the claim

a

x

b

u2 (x1 )

P

u1

u1

u4 (x2 ) u3 (x3 )

x

x3

u2 (x1 ) u4 (x2 ) u3

c

P1

x

u2 (x1 ) u1

x3 u4 (x2 )

u3

P2

96

8 Maximum Generalized Local Connectivity

a

x

b

x

x3 x2

x2

P

P

x3 x1

P

x

x1

x1

c

x3

x2 F1

F2

F3

Fig. 8.6 Graphs for Lemma 8.2.10

the edges in fu1 u2 ; u1 u4 g, and the tree T3 induced by the edges in fu3 u1 ; u3 u2 ; u3 u4 g form three internally disjoint fu1 ; u2 ; u4 g-Steiner trees, and hence  3 .G/  3; see Fig. 8.5b. Suppose that one element in fu2 ; u4 g is an attaching vertex, say u2 . Then there must exist two paths P1 and P2 connecting x3 and u2 in Hri since Hri is 2-connected. Therefore, the tree T1 induced by the edges in fxu2 ; xu4 ; xx3 g, the tree T2 induced by the edges in fu4 u1 ; u1 u2 g [ E.P1 /, and the tree T3 induced by the edges in fu4 u3 ; u3 u2 g [ E.P2 / form three internally disjoint fu2 ; u4 ; x3 g-Steiner trees, and hence  3 .G/  3; see Fig. 8.5c. Now we consider e … E.K4 /. Then e 2 E.Hri / .1  i  7/. We only consider e 2 E.Hr1 /, and for e 2 E.Hri / .2  i  7/, one can also check that G 2 Gn or  3 .G/  3. Since Hr1 D K2 _ .r  2/K1 , we suppose that e 2 E.K2 _ .r  2/K1 / .r  3/. For r  5, G must contain one element in fF1 ; F2 ; F3 g as its subgraph. One can check that  3 .G/  3 by Observation 8.2; see Fig. 8.6. For r D 4, G 2 H83 or G 2 H84 or G 2 H113 or  3 .G/  3. For r D 3, we can obtain G 2 H72 or G 2 H102 or  3 .G/  3 by Lemma 8.2.2 and Observation 8.3. t u Proof of Theorem 8.2.1. We apply induction on n .n  7/. For n D 3; 4, it is easy to see that Gn D fKn g. For n D 5 or n D 6, the assertion holds by Lemmas 8.2.5 and 8.2.7. Suppose that the assertion holds for graphs of order less than n  7. Now we show that the assertion holds for n  7. We claim that ı.G/  3. Otherwise, ı.G/  4. Let G0 be the graph obtained from G by deleting a vertex x such that dG .x/ D ı.G/. Then, 2e.G0 / D 2e.G/  2dG .x/ D 2e.G/  2ı.G/  .n  2/ı.G/  4.n  2/. But, by the induction hypothesis, 2e.G0 /  2Œ2.n1/3 D 4n10, a contradiction. If ı.G/ D 1, then we let G0 be the graph obtained from G by deleting a pendant vertex. By the induction hypothesis, e.G/  e.G0 / C 1  2.n  1/  3 C 1 D 2n  4 < 2n  3. If ı.G/ D 2, then we let G0 be the graph obtained from G by deleting a vertex of degree 2. If e.G0 / < 2.n1/3, then e.G/ D e.G0 /C2 < 2.n1/3C2 D 2n3. If e.G0 / D 2.n  1/  3, then e.G/ D e.G0 / C 2 D 2.n  1/  3 C 2 D 2n  3. Since  G0 2 Gn1 and  3 .G/  2, it follows from Lemma 8.2.8 that G 2 Gn . Suppose now that ı.G/ D 3. Let G0 be the graph obtained from G by deleting a  . We can get a vertex of degree 3, say x. If e.G0 / D 2.n  1/  3, then G0 2 Gn1 0 contradiction by Lemma 8.2.9. If e.G / < 2.n  1/  3, then e.G/ D e.G0 / C 3  2.n  1/  4 C 3 D 2n  3.

8.2 Graphs with at Most Two Internally Disjoint Steiner Trees Connecting Any. . .

97

Now we will show that G 2 Gn for e.G0 / D 2.n  1/  4. Suppose NG .x/ D fx1 ; x2 ; x3 g. Then we have the following two cases to consider: Case 1. GŒNG .x/ is not a triangle. In this case, there exists an edge xi xj … E.G/ .1  i; j  3/. Let G00 D G0 C xi xj . We claim that  3 .G00 /  2. Assume, to the contrary, that  3 .G00 /  3. Then there exists a 3-subset S  V.G/ such that G00Scontains three internally disjoint S-Steiner trees, denoted by T1 ; T2 ; T3 . If xi xj … 3iD1 E.Ti /, then T1 ; T2 ; T3 are 3 internally disjoint S-Steiner trees in G, which contradicts  3 .G/  2. Assume that xi xj belongs to some S-Steiner tree, without loss of generality, say xi xj 2 E.T1 /. Then the tree T10 induced by the edges in .E.T1 / n xi xj / [ fxi x; xxj g is an S-Steiner tree in G. Thus, T10 ; T2 ; T3 are three internally disjoint S-Steiner trees in G, and hence  3 .G/  3, a contradiction.  Since e.G00 / D e.G0 / C 1 D 2.n  1/  3 and  3 .G/  2, we have G00 2 Gn1 .  Furthermore, G 2 Gn by Lemma 8.2.10. Case 2. GŒNG .x/ is a triangle. Note that GŒNG Œx is a clique of order 4, where NG Œx D NG .x/ [ fxg. From Lemma 8.2.2, there is no path connecting any two vertices of GŒNG Œx. So, G n E.GŒNG Œx/ has three connected components except x, denoted by PG1 ; G2 ; G3 . Note that Gi ¤ K4 .i D 1; 2; 3/. By the induction hypothesis, e.G/ D 3iD1 e.Gi / C 6  P 2 3iD1 jV.Gi /j  3 D 2.n  1/  3 < 2n  3. t u By the definition of f .nI  k  `/, the following corollary is immediate. Corollary 8.2.11 ([74]).  f .nI  3  2/ D

2n  2 if n D 4 2n  3 if n  3; n ¤ 4:

For general `, they constructed a graph class to give a lower bound of f .nI  3  `/. Example 8.1. Let n; ` be odd and G0 be a graph obtained from an .`  3/-regular graph of order n  2 by adding a maximum matching and G D G0 _ K2 . Then ı.G/ D `  1,  3 .G/  `, and e.G/ D `C2 .n  2/ C 12 . 2 Otherwise, let G0 be an .`  2/-regular graph of order n  2 and G D G0 _ K2 . Then ı.G/ D `,  3 .G/  ` and e.G/ D `C2 .n  2/ C 1. 2 Therefore, ( f .nI  3  `/ 

`C2 .n 2

 2/ C

1 2

`C2 .n 2

 2/ C 1 otherwise

if n; ` odd

One can see that for ` D 2, this bound is the best possible (f .nI  3  2/ D 2n3). Actually, the graph constructed for this bound is K2 _.n2/K1 , which belongs to Gn .

98

8 Maximum Generalized Local Connectivity

Fig. 8.7 Graphs for f .nI  k D 1/ D1

D6

D2

D3

D7

D4

D8

D5

D9

Li and Zhao [97] investigated the exact value of f .nI  k D 1/ and introduced the following operation and graph class: Let H1 and H2 be two connected graphs. The graph H1 C H2 is obtained from H1 and H2 by joining an edge v1 v2 between H1 and H2 where v1 2 V.H1 / and v2 2 V.H2 /. This operation is called the adding operation. For example, fC3 gi CfC4 gj CfC5 gk CfK1 g` is a class of connected graphs obtained from i copies of C3 , j copies of C4 , k copies of C5 , and ` copies of K1 by some adding operations such that 0  i  b n3 c, 0  j  2, 0  k  1, 0  `  2, and 3i C 4j C 5k C ` D n. Note that these operations are taken in an arbitrary order. The following graphs shown in Fig. 8.7 will be used later. At first, they studied the exact value of f .nI  3 D 1/ and characterized the graphs attaining this value. The graph class Dnq is defined as follows: Let n D 3r C q, where 0  q  2. If q D 0, then Dn0 D fC3 gr . If q D 1, then Dn1 D fC3 gr C K1 or fC3 gr1 C C4 . If q D 2, then Dn2 D fC3 gr C fK1 g2 or fC3 gr1 C C4 C K1 or fC3 gr1 C C5 or fC3 gr2 C fC4 g2 . Theorem 8.2.15 ([97]). Let n D 3r C q .0  q  2/, and let G be a connected graph of order n such that  3 .G/ D 1. Then e.G/ 

4n  3  q 3

with equality if and only if G 2 Dnq . Next, they investigated the exact value of f .nI  4 D 1/ and characterized the graphs attaining this value. The graph class fD1 gi0 C fD2 gi1 C fD3 gi2 C fD4 gi3 C fD5 gi4 C fD6 gi5 C fD7 gi6 C fD8 gi7 C fD9 gi8 C fK1 gi9 is composed of another connected graph class by some adding operations satisfying the following conditions: • 0  i0  2, 0  i1  b n4 c, 0  i2 C i3 C i4  2, 0  i5 C i6 C i7 C i8  1, 0  i9  2; • Di and Fj are not simultaneously in a graph belonging to this graph class where 1  i  3, 1  j  4; • 3i0 C 4i1 C 5.i2 C i3 C i4 / C 6.i5 C i6 C i7 C i8 / C i9 D n. The graph class Fnq is defined as follows: Let n D 4r C q, where 0  q  3. If q D 0, then Fn0 D fD2 gr . If q D 1, then Fn1 D fD2 gr C K1 or fD2 gr1 C Di .3  i  5). If q D 2, then Fn2 D fD2 gr CfK1 g2 or fD2 gr1 CfD1 g2 or fD2 gr1 CDi CK1

8.2 Graphs with at Most Two Internally Disjoint Steiner Trees Connecting Any. . .

99

or fD2 gr2 C Di C Dj .3  i; j  5/ or fD2 gr1 C Di .6  i  9/. If q D 3, then Fn3 D fD2 gr C D1 . Theorem 8.2.16 ([97]). Let n D 4r C q where 0  q  3, and let G be a connected graph of order n such that  4 .G/ D 1. Then 8 3n2 ˆ if q D 0 ˆ 2 ˆ ˆ < 3n3 if q D 1 2 e.G/  3n4 ˆ if q D 2 ˆ 2 ˆ ˆ : 3n3 if q D 3 2 with equality if and only if G 2 Fnq . For a graph G, we say that a path P D u1 u2    uq is an ear of G if V.G/\V.P/ D fu1 ; uq g. If u1 ¤ uq , then P is an open ear; otherwise P is a closed ear. In their proofs of Theorems 8.2.15 and 8.2.16, they got necessary and sufficient conditions for  k .G/ D 1 with k D 3; 4 by means of the number of ears. Naturally, one might think that this method can always be applied for k D 5, i.e., every cycle in G has at most two ears, but unfortunately they found a counterexample. Example 8.2. Let G be a graph which contains a cycle with three independent closed ears. Set C D u1 u2 u3 , P1 D u1 v1 w1 u1 , P2 D u2 v2 w2 u2 , and P3 D u3 v3 w3 u3 . Then  5 .G/ D 1. In fact, let S be the set of chosen five vertices. Obviously, for each i, if vi and wi are in S, then .S/ D 1. So, only one vertex in Pi  ui can be chosen. Suppose that v1 ; v2 ; v3 have been chosen. By symmetry, u1 ; u2 are chosen. It is easy to check that there is only one Steiner tree connecting fu1 ; u2 ; v1 ; v2 ; v3 g. The remaining case is that all u1 , u2 , and u3 are chosen. Then, no matter which are the another two vertices, only one tree can be found. For a general k with 5  k  n  1, they obtained the following lower bound of f .nI  k .G/ D 1/ by constructing a graph  class as follows: If q D 0, then we let C r  1. If 1  q  k  2, then we let G D fKk1 gr , and hence e.G/ D r k1 2  q C 2 C r. So the following proposition G D fKk1 gr C Kq , and hence e.G/ D r k1 2 is immediate. Proposition 8.2.17 ([97]). For n D r.k  1/ C q .0  q  k  2/, ( k1 r 2 C r  1 if q D 0 f .nI  k D 1/    q r k1 C 2 C r if 1  q  k  2 2 For k D n, Li and Zhao also got the exact value of f .nI  k D 1/. The graph class Kn is defined as follows: K5 D fG j jV.G/j D 5; e.G/ D 7g for n D 5; Kn D Kn1 C K1 for n  6. Theorem 8.2.18 ([97]). Let G be a connected graph of order n such that  n .G/ D 1 where n  5. Then

100

8 Maximum Generalized Local Connectivity

! n1 e.G/  C1 2 with equality if and only if G 2 Kn . The following three corollaries are immediate from Theorems 8.2.15, 8.2.16, and 8.2.18: Corollary 8.2.19 ([97]). For n D 3r C q .0  q  2/, f .nI  3 D 1/ D

4n  3  q : 3

Corollary 8.2.20 ([97]). For n D 4r C q .0  q  3/, 8 3n2 ˆ if q D 0 ˆ 2 ˆ ˆ < 3n3 if q D 1 2 f .nI  4 D 1/ D 3n4 ˆ if q D 2 ˆ 2 ˆ ˆ : 3n3 if q D 3 2 Corollary 8.2.21 ([97]). For n  5, f .nI  n D 1/ D

n1 2

C 1.

8.3 Graphs with at Most ` Internally Disjoint Steiner Trees Connecting Any n  1 Vertices Note that for k D n, we have 1  `  b n2 c by Observation 8.1. With the help of Theorem 1.4.1 (due to Nash-Williams and Tutte), Li and Mao determined the exact value of f .nI  k  `/ for k D n. Theorem 8.3.1 ([87]). Let G be a connected graph of order n .n  6/. If n .G/  ` .1  `  b n2 c/, then 8 n1 ˆ C` if 1  `  b n4 c ˆ 2 2 ˆ   ˆ < n1 C n  2 if ` D b n2 c and n is even 2 2 e.G/  n1  n3 n2 ˆ C if ` D b c and n is odd ˆ 2 2 2 ˆ ˆ : n n if ` D b 2 c 2 c where In is a graph class with equality if and only if G 2 In for 1  `  b n4 2 obtained from a complete graph Kn1 by adding a vertex v and joining v to ` vertices of Kn1 , G D Kn n e where e 2 E.Kn / for ` D b n2 c and n even, G D Kn n M where 2 n2 for ` D b c and n odd, and G D Kn for ` D b n2 c. M  E.Kn / and jMj D nC1 2 2

8.3 Graphs with at Most ` Internally Disjoint Steiner Trees Connecting Any. . .

101

From the definitions of f .nI  n  `/ and g.nI n  `/, the following corollary is immediate. Corollary 8.3.2 ([87]). For 1  `  b n2 c and n  6, 8 n1 C ` if 1  `  b n4 c or ` D b n2 c ˆ ˆ 2 2 2 ˆ < and n is odd f .nI  n  `/ D g.nI n  `/ D n1 C 2` if ` D b n2 c and n is even ˆ 2 ˆ ˆ n2 : n if ` D b 2 c 2 Before giving our main results, we need some preliminary results. From Observation 8.1, we know that 1   n1 .G/  b nC1 c. So we only need to consider 2 c. In order to determine the exact value of f .nI  n1  `/ for a general 1  `  b nC1 2 nC1 c/, we first focus on the cases ` D b c and b n1 c. These cases ` .1  `  b nC1 2 2 2 n3 will also be used when dealing with the case ` D b 2 c. c and ` D b n1 c The Subcases ` D b nC1 2 2 The following results can be derived from Lemma 5.1.1. Lemma 8.3.3 ([87]). Let G D Kn n M be a connected graph of order n .n  4/, where M  E.Kn /: (1) If n is odd and jMj  1, then n1 .G/ < nC1 ; 2 (2) If n is even and jMj  n2 , then n1 .G/ < n2 . With the help of Lemma 8.3.3, Theorems 5.1.5 and 5.1.6, and Observation 8.1, the graphs with  n1 .G/ D b nC1 c can now be characterized. 2 Proposition 8.3.4 ([87]). For a connected graph G of order n .n  4/,  n1 .G/ D b nC1 c if and only if G D Kn for n odd or G D Kn n M for n even, where M is a set 2 . of edges such that 0  jMj  n2 2 c can also be characterized. Furthermore, graphs with n1 .G/ D b nC1 2 Proposition 8.3.5 ([87]). For a connected graph G of order n .n  4/, n1 .G/ D c if and only if G D Kn for n odd or G D Kn n M for n even, where M is a set b nC1 2 . of edges such that 0  jMj  n2 2 We now focus our attention on the case ` D b n1 c. Before characterizing the 2 graphs with n1 .G/ D b n1 c, we need the following four lemmas. The notion 2 of a second smallest degree vertex in a graph G will be used in the sequel. If G has two or more minimum degree vertices, then, choosing one of them as the first minimum degree vertex, a second smallest degree vertex is defined as any one of the rest minimum degree vertices of G. If G has only one minimum degree vertex, then a second smallest degree vertex is as its name suggests, defined as any one of vertices that have the second smallest degree. Note that a second smallest degree vertex is usually not unique.

102

8 Maximum Generalized Local Connectivity

Lemma 8.3.6 ([87]). Let G D Kn n M be a connected graph of order n, where M  E.Kn /: (1) If n .n  10/ is even and jMj  3n4 , then n1 .G/ < n1 ; 2 2 3n6 (2) If n .n  10/ is even, nC1  jMj  2 , and there is a second smallest degree vertex, say u1 , such that dG .u1 /  n4 , then n1 .G/ < n2 ; 2 2 n1 (3) If n .n  8/ is odd and jMj  n  1, then n1 .G/ < 2 . Lemma 8.3.7 ([87]). Let H be a connected graph of order n  1:   (1) If n .n  5/ is odd, e.H/  n2 , ı.H/  n3 , and any two vertices of degree 2 2 n3 n3 are nonadjacent, then H contains edge-disjoint spanning trees. 2   n22 n4  , ı.H/  , and any two vertices of (2) If n .n  7/ is even, e.H/  n2 2 2 2 n4 n4 degree 2 are nonadjacent, then H contains 2 edge-disjoint spanning trees. The following theorem, due to Dirac, is well known. Theorem 8.3.8 ([20], p. 485). Let G be a simple graph of order n .n  3/ and minimum degree ı. If ı  n2 , then G is Hamiltonian. Lemma 8.3.9 ([87]). If n  11 is odd and G D Kn n M where jMj D n  2, then  n1 .G/  n1 . 2 Lemma 8.3.10 ([87]). If n  14 is even and G D Kn n M where jMj D 3n6 and 2 n2 for some second smallest degree vertex u in G, then  .G/  . dG .u1 /  n2 1 n2 2 2  n2   n4 Proof. It is clear that e.G/ D 2 C n2 D n1 and ı.G/  n  4. Let 2 2 ı.G/ D r and v be a vertex such that dG .v/ D ı.G/ D r. Let S D V.G/  v D fu1 ; u2 ;    ; un1 g. Without loss of generality, let S1 D fu1 ; u2 ;    ; ur g such that vuj 2 E.G/ .1  j  r/. Then S2 D S  S1 D furC1 ; urC2 ;    ; un1 g such that vuj 2 M .r C 1  j  n  1/. When ı.G/ D n  4, we have ı.GŒS/  n  5. Recall that S1 D fu1 ; u2 ;    ; un4 g such that vuj 2 E.G/ .1  j  n  4/. Thus S2 D S  S1 D fun1 ; un2 ; un3 g and vun1 ; vun2 ; vun3 2 M. Since ı.G/ D n  4 and vui … E.G/ for each ui .i 2 fn  1; n  2; n  3g/, it follows that jEG Œui ; S1 j  n  4  2 D n  6 > 3. Then there exist three vertices in S1 , say ui1 ; ui2 ; ui3 , such that ui1 un1 ; ui2 un2 ; ui3 un3 2 E.GŒS/ (note that it can always be done that ui1 ; ui2 ; ui3 are different). Then the tree T induced by the edges in fvu1 ; vu2 ;    ; vun4 g [ fui1 un1 ; ui2 un2 ; ui3 un3 g is an S-Steiner tree. Set G1 D GnE.T/. Observe that ı.G1 ŒS/  n51 D n6 > n4 . 2 n2 n2 Since e.G1 ŒS/ D e.G/  .n  1/ D 2  2 , it follows from .2/ of Lemma 8.3.7 edge-disjoint spanning trees. These trees together with the that G1 ŒS contains n4 2 internally disjoint S-Steiner trees. Therefore,  n1 .G/  n2 , as tree T result in n2 2 2 desired. The case when ı.G/ D n  5 can be proved similarly. In the following we can assume that 1  ı.G/  n  6. We have two cases to consider:

8.3 Graphs with at Most ` Internally Disjoint Steiner Trees Connecting Any. . . Fig. 8.8 Graphs for Lemma 8.3.10

a

b u3

ur−2 Pr−2

ur−1 Pr−1 ur

Pr

v

Case 1. 1  ı.G/ 

n2 . 2

103

P2 u1

ur u2 P1

S2

uq u1

ut+2 S1 ut+1 ut up

v

  n2  2 , and hence G  v is a graph If dG .v/ D ı.G/ D 1, then e.G  v/ D n1 2 edges. From Corollary 8.1.1, obtained from a clique of order n  1 by deleting n2 2 n2 . Therefore,  .G/  . Suppose ı.G/  2. Since we have  n1 .G  v/ D n2 n1 2 2 n2 n2 n ı.G/  2 , it follows that dKn ŒM .v/  n  1  2 D 2 and hence jM \ Kn ŒSj  where u1 is a second smallest degree vertex, we have n  3. Since dG .u1 /  n2 2 . ı.GŒS/  n4 2 First, we consider the case ı.GŒS/  n2 . We claim that there are at most two vertices of degree n2 in GŒS. Assume, to the contrary, that there are three vertices for j D 1; 2; 3, and hence of degree n2 in GŒS, say u1 ; u2 ; u3 . Then dG .uj /  nC2 P3 2 n n4 . Therefore, jMj  d .v/ C d dKn ŒM .uj /  n4 Kn ŒM jD1 Kn ŒM .uj /  2 C 3  2  2 3 D 2n  9 > 3n6 , a contradiction. From the above, we conclude that there exist 2 , it follows at most two vertices of degree n2 in GŒS. Since ı.GŒS/  n2 > n1 2 from Theorem 8.3.8 that GŒS is Hamiltonian and hence GŒS contains a Hamilton cycle, say C. Then the vertices u1 ; u2 ;    ; ur divide the cycle C into r paths, say P1 ; P2 ;    ; Pr ; see Fig. 8.8a. Now we find an S-Steiner tree with its root v in G, say T, such that G1 ŒS satisfies the conditions of .2/ in Lemma 8.3.7, where G1 D G  E.T/. If there exist two adjacent vertices us ; up 2 S of degree n2 in GŒS such that us ; up 2 fu1 ; u2 ;    ; ur g, then p D s C 1, and Ps D us usC1 .1  s  r  1/. Since dGŒS .us / D n2 and r  n2 , it follows that there exists a vertex ut 2 2 furC1 ; urC2 ;    ; un1 g such that us ut 2 E.G/. It is clear that ut is an internal vertex of some path; without loss of generality, let ut 2 V.Pq / .1  q  r; q ¤ s/. For each path Pi .1  i  r/, we choose one edge ei 2 E.Pi / .1  i  r/ to delete. Since ut is an internal vertex of Pq , it is incident with two edges of Pq , say eq and e0q . Then the tree T induced by the edges in fvu1 ; vu2 ;    ; vur g [ E.P1 n e1 / [ E.P2 n e2 / [    [ E.Pr n er / [ fus ut g n fe0q g is the required tree. Set G1 D G n E.T/. Observe and there is at most one vertex of degree n4 in G1 ŒS. Since that ı.G1 ŒS/  n4 2 2 n2 n2 e.G1 ŒS/ D e.G/  .n  1/ D 2  2 , it follows from .2/ of Lemma 8.3.7 that edge-disjoint spanning trees. These trees together with the tree G1 ŒS contains n4 2 n2 , as desired. T are 2 internally disjoint S-Steiner trees. Thus,  n1 .G/  n2 2 Apart from the above case, we also have the following five cases to consider. For each case, we choose one edge ei 2 E.Pi / .1  i  r/ to delete that satisfies the following conditions:

104

8 Maximum Generalized Local Connectivity

(1) If there is at most one vertex of degree n2 , then ei .1  i  r/ is chosen as any edge in Pi . (2) If there exist two adjacent vertices us ; ut 2 S of degree n2 in GŒS such that us 2 fu1 ; u2 ;    ; ur g and ut 2 furC1 ; urC2 ;    ; un1 g, then es is the edge that is incident with usC1 , and ei .1  i  r; i ¤ s/ is chosen as any edge in Pi . (3) If there exist two adjacent vertices us ; ut 2 S of degree n2 in GŒS such that us 2 furC1 ; urC2 ;    ; un1 g and ut 2 fu1 ; u2 ;    ; ur g, then et1 is the edge that is incident with ut1 , and ei .1  i  r; i ¤ t  1/ is chosen as any edge in Pi . (4) If there exist two adjacent vertices us ; ut 2 S of degree n2 in GŒS such that us ; ut 2 furC1 ; urC2 ;    ; un1 g, then ei .1  i  r/ is chosen as any edge in Pi . (5) If there exist two nonadjacent vertices us ; ut 2 S of degree n1 in GŒS, then 2 ei .1  i  r/ is chosen as any edge in Pi . Then the tree T induced by the edges in fvu1 ; vu2 ;    ; vur g [ E.P1 n e1 / [ E.P2 n e2 / [    [ E.Pr n er / is an S-Steiner tree. Obviously, ı.G1 ŒS/  n4 , and there is 2   n2 n4  2 , at most one vertex of degree 2 . Since e.G1 ŒS/ D e.G/  .n  1/ D n2 2 edge-disjoint spanning it follows from .2/ of Lemma 8.3.7 that G1 ŒS contains n4 2 internally disjoint S-Steiner trees. These trees together with the tree T result in n2 2 , as desired. trees. Therefore,  n1 .G/  n2 2 and ı.GŒS/ D n4 . If ı.GŒS/ D Next, we focus on the case that ı.GŒS/ D n2 2 2 n4 n4 , then there exists a vertex, say u , such that d .u / D . Since the degree 1 GŒS 1 2 2 n2 of a second smallest degree vertex is not less than 2 , we have u1 2 S1 . Thus and u1 2 S1 . If ı.GŒS/ D n2 , then there exists a vertex, say u1 , such dG .u1 / D n2 2 2 n2 n and u 2 S or d .u / that dGŒS .u1 / D n2 1 1 GŒS 1 D 2 and u1 2 S2 . Thus dG .u1 / D 2 2 and u1 2 S1 , or dG .u1 / D n2 and u1 2 S2 . We only give the proof of the case that 2 dG .u1 / D n2 and u1 2 S1 . The other two cases can be proved similarly. Suppose dG .u1 / D n2 and u1 2 S1 . Then it is not difficult to construct a Steiner tree T for S with root v. Let G1 D G n E.T/. We then show that the graph G1 ŒS satisfies the conditions of .2/ in Lemma 8.3.7. This implies that there are n4 edge2 disjoint spanning trees connecting S in G1 ŒS, and these trees together with the tree T are n2 internally disjoint S-Steiner trees. So,  n1 .G/  n2 . Let S10 D S1  u1 2 2 0 0 0 and S D S1 [ S2 . Let us focus on the graph GŒS1 . If r D 2, then GŒS0  is a graph obtained from a clique of order n  2 by deleting one edge since dKn ŒM .u1 / D n2 2 and dKn ŒM .v/ D n  3 and jMj D 3n6 . Without loss of generality, let NG .v/ D 2 fu1 ; u2 g. Clearly, GŒS0  contains a Hamilton path P with u2 as one of its endpoints. Then the tree T induced by the edges in fvu1 ; vu2 g [ E.P/ is our desired tree. Set G1 D G n E.T/. Thus ı.G1 ŒS0 / D ı.GŒS0 /  2  n  4  2 D n  6  n2 . 2 , the result follows by .2/ of Lemma 8.3.7. We now assume Since dG1 ŒS .u1 / D n2 2 , dKn ŒM .v/  n2 , and jMj D 3n6 , GŒS0  is a graph r  3. Since dKn ŒM .u1 / D n2 2 2 n4 obtained from the complete graph Kn2 by deleting at most 2 edges, and hence D n2 . It is clear that there exist at least two vertices of ı.GŒS0 /  n  3  n4 2 2 0 in GŒS0 . degree n  3 in GŒS , and there is also at most one vertex of degree n2 2 Without loss of generality, let ui1 ; ui2 be two vertices of degree n  3.

8.3 Graphs with at Most ` Internally Disjoint Steiner Trees Connecting Any. . .

a S2

u1

ur+1

b

ur+1+ n−4

S2

2

un−1 ur+ n−4 2

u2

ur S1

u1

ur+1

v

c

ur+1+ n−4 un−1 2

un−2 ur+ n−4 u2

2

S2

ur+ n−4 2

S1

v

ur+1

un−1

un−3 un−2 ur+1+ n−4

ur

u1

105

2

u2

ur S1

v

Fig. 8.9 Graphs for case 1 of Lemma 8.3.10

If ui1 ; ui2 2 S10 , without loss of generality, let ui1 D u2 and ui2 D ur ; then the tree T induced by the edges in fvu1 ;    ; vur g [ fu2 urC1 ;    ; u2 urC n4 g [ 2 fur urC n4 C1 ;    ; ur un1 g is an S-Steiner tree; see Fig. 8.9a. Set G1 D G n 2

, dG1 ŒS .u2 /  n  3  n4 D n2 , and E.T/. Observe that dG1 ŒS .u1 / D n2 2 2 2 n4 n4 n2 dG1 ŒS .ur / D .n  3/  .n  1  r  2 / D r  2 C 2  2 . For uj 2 S2 (r C 1  j  n  1), dG1 ŒS .uj /  n4 , and there is at most one vertex of 2 n4 in G ŒS. So ı.G ŒS/  , and there is at most one vertex of degree n4 1 1 2 2 in G ŒS, as desired. If u 2 S10 and ui2 2 S2 , without loss of degree n4 1 i1 2 generality, let ui1 D u2 and ui2 D un1 ; then the tree T induced by the edges in fvu1 ;    ; vur g[fu2 urC1 ;    ; u2 urC n4 g[fun1 urC n4 C1 ;    ; un1 un2 ; un1 ur g is 2

2

the required tree; see Fig. 8.9b. Set G1 D G n E.T/. One can see that ı.G1 ŒS/  n4 2 and there is at most one vertex of degree n4 in G1 ŒS, as desired. Let us consider 2 the remaining case ui1 ; ui2 2 S2 . Without loss of generality, let ui1 D un1 and ui2 D un2 . The tree T induced by the edges in fvu1 ;    ; vur g [ fun2 urC1 ;    ; un2 urC n4 g[fun1 urC n4 C1 ;    ; un1 un3 ; u2 un2 ; un1 ur g is our desired tree; see 2

2

and that there Fig. 8.9c. Set G1 D G n E.T/. One can see that ı.G1 ŒS/  n4 2 in G ŒS. Using .2/ of Lemma 8.3.7, we get is at most one vertex of degree n4 1 2 n2  n1 .G/  2 , as desired. Case 2.

n 2

 ı.G/  n  6.

Recall that S1 D fu1 ; u2 ;    ; ur g with vuj 2 E.G/ .1  j  r/ and S2 D S S1 D furC1 ; urC2 ;    ; un1 g. Obviously, jS1 j D r D ı.G/  n2 and jS2 j D n  1  r  . For each uj 2 S2 .r C 1  j  n  1/, uj has at most n4 neighbors n  1  n2 D n2 2 2 n D 2 since d .u /  ı.G/  . Clearly, in S2 , and hence jEG Œuj ; S1 j  n2  n4 G j 2 2 the tree T 0 induced by the edges in fvu1 ; vu2 ;    ; vur g is an S1 -Steiner tree. Our idea is to find n  1  r edges in EG ŒS1 ; S2  and combine them with T 0 to form an S-Steiner tree. Choose the one with the smallest subscript among all the vertices of S2 having maximum degree in GŒS, say u01 . Then we select the vertex adjacent to u01 with the smallest subscript among all the vertices of S1 having maximum degree in GŒS, say u001 . Let e1 D u01 u001 . Consider the graph G1 D G n e1 , and choose the vertex with the smallest subscript among all the vertices of S2 u01 having maximum degree in G1 ŒS, say u02 . Then we select the vertex adjacent to u02 with the smallest

106

8 Maximum Generalized Local Connectivity

subscript among all the vertices of S1 having maximum degree in G1 ŒS, say u002 . Set e2 D u02 u002 . We consider the graph G2 D G1 n e1 D G n fe1 ; e2 g. For each uj 2 S2 .r C 1  j  n  1/, we proceed to find e3 ; e4 ;    ; en1r in the same way. Let M 0 D fe1 ; e2 ;    ; en1r g and Gn1r D G n M 0 . Then Gn1r ŒS D GŒS n M 0 , and the tree T induced by the edges in fvu1 ; vu2 ;    ; vur g [ fe1 ; e2 ;    ; en1r g is the required tree. Set G0 D G n E.T/ (note that G0 ŒS D Gn1r ŒS). Claim 1. Any uj 2 S1 .1  j  r/ has a degree dG0 ŒS .uj /  n4 , and if there exist 2 n4 0 vertices with degree equal to 2 in G ŒS, then the number of such vertices is at most two, and they are nonadjacent. Proof of Claim 1. First, we prove that any uj 2 S1 .1  j  r/ has a degree dG0 ŒS .uj /  n4 . Assume, to the contrary, that there exists a vertex up 2 S1 such 2 0 . By the above procedure, there exists a vertex uq 2 S2 such that dG ŒS .up /  n6 2 that when we select the edge ei D up uq from Gi1 ŒS, the degree of up in Gi ŒS is equal to n6 . Thus dGi ŒS .up / D n6 and dGi1 ŒS .up / D n4 . From our procedure, 2 2 2 jEG Œuq ; S1 j D jEGi1 Œuq ; S1 j. Without loss of generality, let jEG Œuq ; S1 j D t and uq uj 2 E.G/ for 1  j  t; see Fig. 8.8b. Thus up 2 fu1 ; u2 ;    ; ut g. Recall that jEG Œuj ; S1 j  2 for each uj 2 S2 .r C 1  j  n  1/. Since uq 2 S2 , we have t  2. Observe that uq uj … E.G/ and hence uq uj 2 M for t C 1  j  r by our procedure, namely, jEKn ŒM Œuq ; S1 j D r  t. Since dGi1 ŒS .up / D n4 , by 2 for each u 2 S .1  j  t/. Assume, to the our procedure dGi1 ŒS .uj /  n4 j 1 2 . Then contrary, that there is a vertex us .1  s  t/ such that dGi1 ŒS .us /  n2 2 we would have selected the edge uq us instead of ei D uq up by our procedure, a contradiction. We conclude that dGi1 ŒS .uj /  n4 for each uj 2 S1 .1  j  t/. 2 n4 Clearly, there are at least n  2  2 edges incident to each uj .1  j  t/ that belong to M [ fe1 ; e2 ;    ; ei1 g. Since i  n  1  r, we have  nt n  4 t  .i  1/   .n  2  r/ dKn ŒM .uj /  n  2  2 2 jD1

t X

and hence ! t jMj  dKn ŒM .v/ C dKn ŒM .uj /  C jEKn ŒM Œuq ; S1 j 2 jD1 t X

nt .t  1/t  .n  2  r/  C .r  t/ 2 2  t2 .n  1/t n n  C C C1 since r  2 2 2 2 3n  4  .since 2  t  n  6/; 2  .n  1  r/ C

which contradicts the fact that jMj D

3n6 . 2

8.3 Graphs with at Most ` Internally Disjoint Steiner Trees Connecting Any. . .

107

Next, we prove that if there exist vertices with degree equal to n4 in G0 ŒS, then 2 the number of such vertices is at most two, and they are nonadjacent. Assume, to the contrary, that there exist at least three vertices of degree n4 in G0 ŒS or there 2 in G0 ŒS, say up0 ; up . We only prove the exist two adjacent vertices of degree n4 2 latter case; the former case can be proved similarly. By the above procedure, there exists a vertex uq0 2 S2 such that when we select the edge ei0 D up0 uq0 from Gi0 1 ŒS, the degree of up in Gi0 ŒS equals n4 . Thus dGi0 ŒS .up0 / D n4 . By the same reason, 2 2 there exists a vertex uq 2 S2 such that when we select the edge ei D up uq from Gi1 ŒS, the degree of up in Gi ŒS is equal to n4 , that is, dGi ŒS .up / D n4 and 2 2 n2 dGi1 ŒS .up / D 2 . Without loss of generality, let i0 < i. From our procedure, jEG Œuq ; S1 j D jEGi1 Œuq ; S1 j. Without loss of generality, let jEG Œuq ; S1 j D t and uq uj 2 E.G/ for 1  j  t; see Fig. 8.8b. Thus up 2 fu1 ; u2 ;    ; ut g. Recall that jEG Œuj ; S1 j  2 for each uj 2 S2 .r C 1  j  n  1/. Since uq 2 S2 , we have t  2. Then uq uj … E.G/, and hence uq uj 2 M for t C 1  j  r by our procedure, namely, jEKn ŒM Œuq ; S1 j D r  t. Since dGi1 ŒS .up / D n2 , by 2 for each u 2 S .1  j  t/. Assume, to the our procedure dGi1 ŒS .uj /  n2 j 1 2 contrary, that there is a vertex us .1  s  t/ such that dGi1 ŒS .us /  n2 . Then, by our procedure, we would have selected the edge uq us instead of ei D uq up , a contradiction. We conclude that dGi1 ŒS .uj /  n2 for each uj 2 S1 .1  j  t/. 2 Suppose up0 2 fu1 ; u2 ;    ; ut g. Without loss of generality, let up0 D u1 . In this case, u1 up … M [ fe1 ; e2 ;    ; en1r g since u1 up 2 E.G0 ŒS/. Then dKn ŒM .u1 / C

t X

dKn ŒM .uj /

jD2

   n  2 .t  1/  .i  1/  n  2  dGi1 ŒS .u1 / C n  2  2   n2 D n  2  dGi0 ŒS .u1 / C .t  1/  .n  2  r/ .since i  n  1  r/ 2   n  4 n  2 n  4  n2 C .t  1/  .n  2  r/ since dGi0 ŒS .u1 /  2 2 2 .n  2/t nC3Cr D 2 and hence ! t jMj  dKn ŒM .v/ C dKn ŒM .u1 / C dKn ŒM .uj /  C 1 C jEKn ŒM Œuq ; S1 j 2 jD2 t X

D .n  1  r/ C 2



 .n  2/t 2

.n  3/t t C CrC3 2 2

 .t  1/t nC3Cr  C 1 C .r  t/ 2

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8 Maximum Generalized Local Connectivity

.n  3/t n n t2 C C C 3 .since r  / 2 2 2 2 3n  4  .since 2  t  n  6/; 2



. which contradicts the fact that jMj D 3n6 2 Suppose up0 … fu1 ; u2 ;    ; ut g. Since up0 up 2 E.G0 ŒS/, it follows that up0 up … M [ fe1 ; e2 ;    ; en1r g. Then up0 2 futC1 ; utC2 ;    ; ur g, and dKn ŒM .up0 / C

t X

dKn ŒM .uj /

jD1

   n  2 t  .i  1/  n  2  dGi1 ŒS .up0 / C n  2  2   n2  n  2  dGi0 ŒS .up0 / C t  .n  2  r/ .since i  n  1  r/ 2  n2 n  4  .t C 1/  n C 3 C r since dGi0 ŒS .up0 /  : 2 2 Furthermore, we have ! tC1 jMj  dKn ŒM .v/ C dKn ŒM .up0 / C dKn ŒM .uj /  C 1 C jEKn ŒM Œuq ; S1 j 2 jD1 p X

 .n  1  r/ C D

n2 .t C 1/t .t C 1/  n C 3 C r  C 1 C .r  t/ 2 2

t2 .n  5/t n2 C CrC C 3: 2 2 2 2

Suppose 2  t  n  7. Since r  n2 , it follows that jMj   t2 C .n5/t C n2 C 2 n2 C 3  2n  5, which contradicts the fact that jMj D 3n6 . If t D n  6, then 2 2 2

r  t D n  6, and hence jMj   .n6/ C .n5/.n6/ C .n  6/ C n2 C 3 D 2n  7, 2 2 2 3n6 t which contradicts the fact that jMj D 2 . The proof of this claim is complete. u for each u 2 S .1  j  r/, and there exists From Claim 1, dG0 ŒS .uj /  n4 j 1 2 in G0 ŒS. For each uj 2 S2 .r C 1  j  n  1/, at most one vertex of degree n4 2 . So ı.G0 ŒS/  n4 , and dG0 ŒS .uj / D dGŒS .uj /  1 D dG .uj /  1  ı.G/  1  n2 2 2 n4 0 there exists at most one vertex of degree 2 in G ŒS. Since e.G0 ŒS/ D e.G/  .n    n2  2 , it follows from .2/ of Lemma 8.3.7 that G0 ŒS contains n4 edge1/ D n2 2 2 internally disjoint spanning trees. These trees together with the tree T result in n2 2 , as desired. t u disjoint S-Steiner trees. Thus,  n1 .G/  n2 2

8.3 Graphs with at Most ` Internally Disjoint Steiner Trees Connecting Any. . .

109

We are now in a position to characterize the graphs with  n1 .G/ D b n1 c. 2 Proposition 8.3.11 ([87]). For a connected graph G of order n .n  13/,  n1 .G/ D b n1 c if and only if G D Kn n M where M  E.Kn / satisfies one 2 of the following conditions: • 1  jMj  n  2 for n odd; • n2  jMj  n for n even; , and dG .u1 /  • n C 1  jMj  3n6 2 vertex in G for n even.

n2 2

where u1 is a second smallest degree

Furthermore, graphs with n1 .G/ D b n1 c can now be characterized. 2 Proposition 8.3.12 ([87]). For a connected graph G of order n .n  13/, n1 .G/ D b n1 c if and only if G D Kn n M where M  E.Kn / satisfies one 2 of the following conditions: • 1  jMj  n  2 for n odd; • n2  jMj  n for n even; , and dG .u1 /  • n C 1  jMj  3n6 2 vertex in G for n even.

n2 2

where u1 is a second smallest degree

The Subcase 1  `  b n5 c 2 c. Now we consider the case 1  `  b n5 2 Lemma8.3.13 graph of order n  1 .n  12/. If  ([87]). Let H be a connected n5 e.H/  n2 C 2`  .n  1/ .1  `  b c/ where ı.H/  ` and any two vertices 2 2 of degree ` are nonadjacent, then H contains ` edge-disjoint spanning trees. Lemma Let G be a connected graph of order n .n  12/. If e.G/  n2 8.3.14 ([87]). n5 C 2` .1  `  b c/, ı.G/  ` C 1, and any two vertices of degree ` C 1 2 2 are nonadjacent, then  n1 .G/  ` C 1. Let Jn be a graph class obtained from the complete graph of order n  2 by adding two nonadjacent vertices and joining each of them to any ` vertices of Kn2 . The following theorem summarizes the results for general `. Theorem 8.3.15 ([87]). Let G be a connected graph of order n .n  13/. If  n1 .G/  ` .1  `  b nC1 c/, then 2 8 n2 C 2` ˆ 2 ˆ ˆ n2  ˆ ˆ ˆ Cn2 ˆ 2 ˆ ˆ < n2 C n  4 2 e.G/  n1  ˆ ˆ 2 ˆ  C n  2 ˆ ˆ n1 ˆ C n2 ˆ ˆ 2 2 ˆ   : n 2

if 1  `  b n5 c 2 if ` D b n3 c and n is odd 2 if ` D b n3 c and n is even 2 if ` D b n1 c and n is odd 2 if ` D b n1 c and n is even 2 if ` D b nC1 c 2

c, G D Kn n M where with equality if and only if G 2 Jn for 1  `  b n5 2 c and n odd, G 2 J for ` D b n3 c and n even, jMj D n  1 for ` D b n3 n 2 2

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8 Maximum Generalized Local Connectivity

G D Kn n e where e 2 E.Kn / for ` D b n1 c and n odd, G D Kn n M where jMj D n2 2 for ` D b n1 c and n even, and G D Kn for ` D b nC1 c. 2 2   n5 Proof. For 1  `  b 2 c, we assume that e.G/  n2 C 2` C 1. Then the 2 following claim is immediate. Claim 1. ı.G/  ` C 1. Proof of Claim 1. Assume, to the contrary, that ı.G/  `. Then there exists a vertex 2 V.G/ such that dG .v/ D ı.G/  `, and hence e.G  v/  e.G/  `  vn2 C ` C 1. Since 1  `  b n5 c, it follows that  n1 .G  v/  ` C 1 by 2 2 Theorem 8.3.1, which results in  n1 .G/  ` C 1, a contradiction. t u From Claim 1, ı.G/  ` C 1. If any two vertices of degree ` C 1 are nonadjacent, then  n1 .G/  ` C 1 by Lemma 8.3.14, a contradiction. So we assume that v1 and v2 are two vertices of degree ` C 1 such that v1 v2 2 E.G/. Let G1 D G  fv1 ; v2 g , and and V.G1 / D fu1 ; u2 ;    ; un2 g. Then e.G1 /  e.G/  .2` C 1/ D n2 2 c  ` C1 hence G1 is a clique of order n  2. Furthermore, G1 contains b n2 2 0 edge-disjoint spanning trees, say T10 ; T20 ;    ; T`C1 . Without loss of generality, let NG .v1 / D fu1 ; u2 ;    ; u` ; v2 g. Choose S D fu1 ; u2 ;    ; un2 ; v2 g. Let Ti be the tree obtained from Ti0 by adding the vertex v1 and the edge v1 ui for i .1  i  `/. Then the trees Ti and Tj are internally disjoint, where 1  i; j  `. Let T`C1 be the 0 tree obtained from T`C1 by adding the vertices v1 ; v2 and the edges v1 v2 ; v2 ut . Then the trees T1 ; T2 ;    ; T`C1 are ` C 1 internally disjoint   S-Steiner trees, and hence C 2` for 1  `  b n5  n1 .G/  ` C 1, a contradiction. So e.G/  n2 c. 2 2 n2 n3 From Proposition 8.3.11, e.G/  2 C n  2 for ` D b 2 c and n odd, and   C n  4 for ` D b n3 c and n even. From Proposition 8.3.4, e.G/  e.G/  n2 2 2 n1   n2 n1 C n  2 for ` D b 2 c and n odd, and e.G/  n1 C 2 for ` D b n1 c and 2 2 2 nC1 n even. If ` D b 2 c, then for any connected graph G, it follows that  n1 .G/  `  by Observation 8.1 and hence e.G/  n2 . c, if Now we characterize the graphs attaining these upper bounds. For ` D b nC1 2 n1  C n  2, then c and n odd, if e.G/ D e.G/ D n2 , then G D Kn . For ` D b n1 2 2  n2 C 2 , then c and n even, if e.G/ D n1 G D Kn ne where e 2 E.Kn /. For ` D b n1 2 2 n2 n n3 G D Kn n M where jMj D 2 . For ` D b 2 c and n odd, if e.G/ D 2 C n  2, then   C n  4 for ` D b n3 c G D Kn n M where jMj D n  1. Suppose that e.G/ D n2 2 2 and n even. From Proposition 8.3.11, G is a graph obtained from a complete graph Kn2 by adding two nonadjacent vertices and adding n4 edges between each of 2 them and the complete graph Kn2 , that is, G 2 Jn .   C 2`. c. Suppose e.G/ D n2 Let us now focus on the case 1  `  b n5 2 2 Similarly to the proof of Claim 1, we can get ı.G/  `. Furthermore, we prove that ı.G/ D `. If ı.G/  ` C 2 or ı.G/ D ` C 1 and any two vertices of degree ` C 1 are nonadjacent, then  n1 .G/  ` C 1 by Lemma 8.3.14, a contradiction. If there exist two vertices of degree ` C 1, say v1 and v2 , such that v1 v2 2 E.G/, then G1 D G  fv1 ; v2 g is a graph obtained from a complete graph of order n  2 by deleting an edge. For n odd, from Corollary 8.1.1 we have  n2 .G1 / D b n2 cD 2 n3 n5 n5  `C1 since `  b 2 c D 2 . For n even, it follows from Corollary 8.1.1 that 2

8.3 Graphs with at Most ` Internally Disjoint Steiner Trees Connecting Any. . .

111

 n2 .G1 /  b n2 c  1 D n4  ` C 1 since `  b n5 c D n6 . We conclude that 2 2 2 2 0 0 G1 contains ` C 1 edge-disjoint spanning trees, say T1 ; T20 ;    ; T`C1 . Set NG .v1 / D 0 fu1 ; u2 ;    ; u` ; v2 g. Let Ti be the tree obtained from Ti by adding the vertex v1 and the edge v1 ui for i .1  i  `/. Then the trees Ti and Tj are internally disjoint, where 0 1  i; j  `. Let T`C1 be the tree obtained from T`C1 by adding the vertices v1 ; v2 and the edges v1 v2 ; v2 ut . Then the trees T1 ; T2 ;    ; T`C1 are ` C 1 internally disjoint S-Steiner trees, which implies that  n1 .G/  ` C 1, a contradiction. So ı.G/ D `. If there exist two vertices of degree `, say v1 ; v2 , such that v1 v 2 2 E.G/, then we set C 1, a contradiction. G1 D G  fv1 ; v2 g. Thus jV.G1 /j D n  2 and e.G1 / D n2 2 So we assume that any two vertices of degree ` are nonadjacent in G. Let p be the number of vertices of degree `. The following claim can easily be proved. Claim 2. 2  p  3.

  P Proof of Claim 2. Assume p  4. Then 2 n2 C 4` D 2e.G/ D v2V.G/ d.v/  2 . Since p  4, it follows that 4n4`4  p`C.np/.n1/, and hence p  4n4`6 n`1 4n  4`  6, a contradiction. Assume p D 1, that is, G contains exactly one  vertex C `. of degree `, say v1 . Set G1 D G  v1 . Clearly, e.G1 / D e.G/  ` D n2 2 Since  n1 .G/  `, it follows that  n1 .G1 /   n1 .G/  `. From Theorem 8.3.1, G1 is a graph obtained from a clique of order n  2 by adding a vertex of degree `, say v2 . Since p D 1, we have dG .v1 / D `, dG .v2 / D ` C 1, and v1 v2 2 E.G/. Observe that G2 D G  fv1 ; v2 g is a clique of order n  2. Thus G2 contains b n2 c 2 0 ` C 1 edge-disjoint spanning trees, say T10 ; T20 ;    ; T`C1 . Without loss of generality, let NG .v1 / D fv2 ; u1 ; u2 ;    ; u` g. Let Ti be the tree obtained from Ti0 by adding the vertex v1 and the edge v1 ui for i .1  i  `/. Then the trees Ti and Tj are 0 internally disjoint, where 1  i; j  `. Let T`C1 be the tree obtained from T`C1 by adding the vertices v1 ; v2 and the edges v1 v2 ; v2 ut . Then the trees T1 ; T2 ;    ; T`C1 are ` C 1 internally disjoint S-Steiner trees, which implies that  n1 .G/  ` C 1, a contradiction. t u From Claim 2, we know that p D 2; 3. If p D 3, then G contains three vertices of degree`, say fv1 ; v2 ; v3 g. Then jV.G1 /j D n  3, and  v1 ; v2 ; v3 . Set n2G1 D G n3 C 2`  3` D  ` > since 1  `  b n5 e.G1 / D n2 c, a contradiction. 2 2 2 2 If p D 2, then G contains two vertices of degree `, say v1 ; v2 . Set G1 D Gfv  1; v2 g. and Since v1 and v2 are nonadjacent, it follows that e.G1 / D e.G/  2` D n2 2 hence G1 is a complete graph of order n  2, which implies that G 2 Jn . t u The following corollary is immediate from Theorem 8.3.15. Corollary 8.3.16 ([87]). For 1  `  b nC1 c and n  13, 2

f .nI  n1

8 n2 C 2`; ˆ 2 ˆ  ˆ n2 ˆ ˆ ˆ <  2  C 2` C 1 n1  `/ D C` 2 ˆ  ˆ ˆ n1 C 2`  1 ˆ ˆ ˆ : n2 2

if 1  `  b n5 c or ` D b n3 c and n is even 2 2 if ` D b n3 c and n is odd 2 if ` D b n1 c and n is even 2 if ` D b n1 c and n is odd 2 if ` D b nC1 c 2

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8 Maximum Generalized Local Connectivity

Applying Theorem 8.3.15 and the relation between  k and k , Li and Mao also investigated the edge case and derived the following result. Theorem 8.3.17 ([87]). Let G be a connected graph of order n .n  13/. If n1 .G/  ` .1  `  b nC1 c/, then 2 8 n2 C 2`; ˆ 2 ˆ ˆ n2  ˆ ˆ Cn2 ˆ ˆ 2  < n2 Cn4 2  e.G/  n1 ˆ Cn2 ˆ 2  ˆ ˆ n1 ˆ C n2 ˆ ˆ 2 : n2 2

if 1  `  b n5 c 2 if ` D b n3 c and n is odd 2 n3 if ` D b 2 c and n is even if ` D b n1 c and n is odd 2 if ` D b n1 c and n is even 2 if ` D b nC1 c 2

c, G D Kn n M where with equality if and only if G 2 Jn for 1  `  b n5 2 c and n odd, G 2 J for ` D b n3 c and n even, jMj D n  1 for ` D b n3 n 2 2 n1 G D Kn n e where e 2 E.Kn / for ` D b 2 c and n odd, G D Kn n M where jMj D n2 for ` D b n1 c and n even, and G D Kn for ` D b nC1 c. 2 2 c/, it follows that  n1 .G/  n1 .G/  ` Proof. Since n1 .G/  ` .1  `  b n5 2     C 2` by Theorem 8.3.15. Suppose e.G/ D n2 C 2`. and hence e.G/  n2 2 2 Since  n1 .G/  n1 .G/  `, we have G 2 Jn by Theorem 8.3.15. For ` D b nC1 c, b n1 c, and b n3 c, respectively, the conclusion holds by Propositions 8.3.5 2 2 2 and 8.3.12. t u Corollary 8.3.18 ([87]). For 1  `  b nC1 c and n  13, 2

g.nI n1

8  n2 ˆ C 2` ˆ 2   ˆ ˆ n2 ˆ <  2  C 2` C 1 n1  `/ D C` 2  ˆ n1 ˆ ˆ C 2`  1 ˆ ˆ : n2 2

if 1  `  b n5 c or ` D b n3 c and n is even 2 2 n3 if ` D b 2 c and n is odd if ` D b n1 c and n is even 2 if ` D b n1 c and n is odd 2 nC1 if ` D b 2 c

Remark 8.3.19 ([87]). What appears to be a difficult problem is to determine the exact value of f .nI  k  `/ and g.nI k  `/ for general k. We now describe examples of graphs that give a lower bound on these parameters. Construct a graph G of order n as follows: Choose a complete graph Kk1 .1  `  b k1 c/. 2 For the remaining n  k C 1 vertices, join each of them k1 to any ` vertices of Kk1 . Clearly,  n1 .G/  n1 .G/  ` and e.G/ D 2 C .n  k C 1/`. So     f .nI  k  `/  k1 C .n  k C 1/`, and g.nI k  `/  k1 C .n  k C 1/`. 2 2 From Theorems 8.3.15 and 8.3.17, one knows that these two bounds are sharp for k D n; n  1.

Chapter 9

Generalized Connectivity for Random Graphs

In this chapter, we introduce results for random graphs. The two most frequently occurring probability models of random graphs are G.n; M/ and G.n; p/. The first consists of all graphs with n vertices having M edges, in which the graphs have the same probability. The model G.n; p/ consists of all graphs with n vertices in which the edges are chosen independently and with probability p. Given sequences an and bn of real numbers (possibly taking negative values), we write: an an an an an an

D O.bn / if there is a constant C > 0 such that jan j  Cjbn j for all n; D o.bn / if eventually bn ¤ 0 and limn!1 an =bn D 0; D ˝.bn / if eventually an  0 and bn D O.an /; D !.bn / if eventually an  0 and bn D o.an /; D .bn / if an  0, an D ˝.bn / and an D .bn /; bn if an D .1 C o.1//bn .

We say that an event A happens almost surely if the probability that it happens approaches 1 as n ! 1, i.e., Pr ŒA  D 1  on .1/. Sometimes, we say a.s. for short. We will always assume that n is the variable that tends to infinity. Given a sequence of events .En /n2N , we say that En happens asymptotically almost surely (a.a.s.) if Pr.En / ! 1 as n ! 1. For a graph property P, a function p.n/ is called a threshold function of P if: • For every r.n/ D O.p.n//, G.n; r.n// almost surely satisfies P ; and • For every r0 .n/ D o.p.n//, G.n; r0 .n// almost surely does not satisfy P. Furthermore, p.n/ is called a sharp threshold function of P if there exist two positive constants c and C such that: • For every r.n/  C  p.n/, G.n; r.n// almost surely satisfies P ; • For every r0 .n/  c  p.n/, G.n; r0 .n// almost surely does not satisfy P.

© The Author(s) 2016 X. Li, Y. Mao, Generalized Connectivity of Graphs, SpringerBriefs in Mathematics, DOI 10.1007/978-3-319-33828-6_9

113

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9 Generalized Connectivity for Random Graphs

The average value (or mean) of a random variable X is called the expectation of X and is denoted by E.X/. The following is a version of the well-known Chernoff’s bounds. TheoremP9.0.1 ([20]). Let X1 ; : : : ; Xn denote n independent Bernoulli variables. Let X D niD1 Xi and let  D EX. Then for any 0 <  < 1, Pr.X  .1 C  //  exp. 2 =3/; Pr.X  .1  //  exp. 2 =2/:

9.1 Results for the Spanning-Tree Packing Number This section is to deal with the case for k D n, i.e., the spanning-tree packing problem for random graphs. Frieze and Łuczak [46] first considered the maximum number of edge-disjoint spanning trees contained in the random graph Gk-out , which has vertex set V D f1; 2;    ; ng, and each v 2 Vn independently chooses a set out.v/ of distinct vertices as neighbors, where each k-subset of Vn  fvg is equally likely to be chosen. This produces a random k out-regular diagraph Dk , which has been selected uniformly from . n1 /n distinct possibilities, where Gk is obtained by k ignoring orientation but without coalescing edges; see [42, 45] for properties of this model. They obtained that for a fixed integer k  2, the random graph Gk-out almost surely has k edge-disjoint spanning trees. Moreover, Palmer and Spencer [115] proved that in almost every random graph process, the hitting time for having k edge-disjoint spanning trees equals the hitting time for having minimum degree k, for any fixed positive integer k. In other words, considering the random graph G.n; p/, for any fixed positive integer k, if p.n/  log nCk lognlog n!.1/ (which is the maximal p for which ı.G.n; p//  k a.s.), the probability that the spanning-tree packing number equals the minimum degree approaches to 1 as n ! 1. On the other hand, in Catlin’s paper [25], it was found that if the edge probability was rather large, then almost surely the random graph G.n; p/ has n .G/ D bjE.G/j=.n1/c, which is less than the minimum degree of G. We refer papers [25] and [114] to the reader for more details. A natural question is whether there exists a largest p.n/ such that for every r0 .n/  p.n/, almost surely the random graph G.n; p/ satisfies that the spanningtree packing number equals the minimum degree. In [31], Chen, Li, and Lian partly answered this question by establishing two results for multigraphs and proved that log n=n is a sharp threshold function for the graph property that the spanning-tree packing number is equal to the minimum degree. Gao, Pérez-Giménez, and Sato [47] strengthened the previous results. In order to introduce their work, we first need more notations and concepts. Let d.G/ D 2m.G/=.jV.G/j  1/. Note that d.G/ differs from the average degree of G by a small factor of jV.G/j=.jV.G/j  1/. Also, let t.G/ D minfı.G/; d.G/=2g.

9.1 Results for the Spanning-Tree Packing Number

115

9.1.1 Main Results Given a graph G, let n.G/, m.G/, and ı.G/ denote the number of vertices, the number of edges, and the minimum degree of G, respectively. Recall that n .G/ is the spanning-tree packing number of G, i.e., the maximum number of edge-disjoint spanning trees in G (possibly 0 if G is disconnected). Gao, Pérez-Giménez, and Sato first proved that for all p.n/ 2 Œ0; 1, the spanningtree packing number is a.a.s. the minimum between ı and bm=.n  1/c. Theorem 9.1.1 ([47]). For every p D p.n/ 2 Œ0; 1, we have that a:a:s:  n .G.n; p// D min ı.G.n; p//;

j m.G.n; p// k n1

:

Note that the quantities ı and bm=.n  1/c above correspond to the two trivial upper bounds observed earlier for arbitrary graphs, so this implies that we can a.a.s. find a best-possible number of edge-disjoint spanning trees in G.n; p/. Their argument uses several properties of G.n; p/ in order to bound the number of crossing edges between subsets of vertices with certain restrictions and then applies the characterization of the spanning-tree packing number by Tutte and Nash-Williams stated in Theorem 1.4.1. Moreover, they determined the ranges of p for which the spanning-tree packing number takes each of these two values: ı and bm=.n  1/c. In spite of the fact that the property fı  bm=.n  1/cg is not necessarily monotonic with respect to p, they showed that it has a sharp threshold at p ˇ log n=n, where ˇ 6:51778 is a constant defined in the following theorem. Theorem 9.1.2 ([47]). Let ˇ D 2= log.e=2/ 6:51778. Then:

˘  log n=2/!.1/ (i) If p D ˇ.log nlogn1 , and so , then a:a:s ı.G.n; p//  m.G.n;p// n1 n .G.n; p// D ı.G.n; p//; ˘  log n=2/C!.1/ (ii) If p D ˇ.log nlogn1 , and so , then a:a:s ı.G.n; p// > m.G.n;p// 2  m.G.n;p// ˘ . n .G.n; p// D 2 Below this threshold, the spanning-tree packing number of G.n; p/ is a:a:s: equal to ı, and above the threshold it is a:a:s: bm=.n  1/c. In particular, this settles the question raised by Chen, Li, and Lian [31]. Gao, Pérez-Giménez, and Sato further considered the random graph process  G0 ; G1 ;    ; G.n/ defined as follows: for each m D 0;    ; n2 , Gm is a graph with 2  vertex set Œn; the graph G0 has no edges; and, for each 1  m  n2 , the graph Gm is obtained by adding one new edge to Gm1 chosen uniformly at random among the edges not present in Gm1 . Equivalently, we can choose uniformly at random a permutation .e1 ;    ; e.n/ / of the edges of the complete graph with vertex set 2 Œn WD f1; 2;    ; ng and define each Gm to be the graph on vertex set Œn and edges e1 ;    ; em . They also included a stronger version of these results in the context of the random graph process in which p gradually grows from 0 to 1 (or, similarly, the edges are

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added one by one). This provides a full characterization of the spanning-tree packing number that holds a:a:s: simultaneously during the whole random graph process. Theorem 9.1.3 ([47]). Let ˇ D 2= log.e=2/ 6:51778. The following holds in the random graph process G0 ; G1 ;    ; G.n/ : 2  (i) a:a:s n .Gm / D minfı.Gm /; bm=.n  1/cg for every 0  m  n2 . (ii) Moreover, for any constant > 0, a:a:s: • ı.Gm /  bm=.n  1/c for every 0  m  .1 /ˇ n log n, 2  .1 /ˇ • ı.Gm / > bm=.n  1/c for every 2 n log n  m  n2 .

9.1.2 Deterministic Tools Theorem 1.4.1 provides a useful characterization of the spanning-tree packing number of a graph G. The next two propositions play a central role in this section. Gao, Pérez-Giménez, and Sato used Theorem 1.4.1 to determine the spanning-tree packing number of any well-behaved graph satisfying certain conditions. For > 0, we say that a vertex of G is -light if its degree is at most ı.G/ C d.G/. Proposition 9.1.4 ([47]). Let G D Gn be a graph on vertex set Œn. Let ı WD ı.G/ and let d WD d.G/. Suppose that d ! 1 as n ! 1 and that there exist constants

; ; > 0 such that the following hold, for all sufficiently large n: (a) The minimum degree ı is at most . =4/d; there is no pair of adjacent -light vertices; and all vertices of G have at most one -light neighbor. (b) No set of size s < n induces more than . =4/ds edges. (c) For all disjoint S; S0  Œn with jSj  jS0 j  n, we have that jEG ŒS; S0 j  dn. Then eventually n .G/ D ı. Proposition 9.1.5 ([47]). Let G D Gn be a graph on vertex set Œn. Let ı WD ı.G/ and d WD d.G/, and suppose that d ! 1 as n ! 1. Let t D minfı; d=2g. Suppose moreover that there exist constants 0 < ; ;  1 such that the following hold, for sufficiently large n: or there are no adjacent -light vertices and (a0 ) Either we have that ı > .1C /d 2 each vertex of G is adjacent to at most one -light vertex. (b0 ) For all S  V.G/, with jSj  n, we have that d.S/  d.1  o.1//, where d.S/ denotes the sum of degrees of vertices in S divided by jSj. (c0 ) For all disjoint S; S0  V.G/ with jSj  jS0 j  n, we have that jEG ŒS; S0 j  djSjjS0 j=n. 0 N  t. (d ) For all ;   S   V.G/, we have that jEG ŒS; Sj (e0 ) No set of size s < n induces more than . =4/ts edges. Then eventually n .G/ D btc.

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Propositions 9.1.4 and 9.1.5 will be used to determine the spanning-tree packing number of G.n; p/. Basically, according to the range of p, they showed that G.n; p/ (or some modification of G.n; p/) satisfies the conditions in Proposition 9.1.4 or Proposition 9.1.5 with sufficiently high probability. Proposition 9.1.4 is applied when the minimum degree is relatively small compared to d, whereas otherwise Proposition 9.1.5 is used instead. Thus, a good estimation of ı.G.n; p// is needed, together with several graph-expansion-related properties, as required by conditions .b/, .c/, .c0 /, .d0 /, and .e0 /, and also some properties about the -light vertices addressed in conditions .a/ and .a0 /. In the following subsection, they gave bounds on the probability that these properties hold in G.n; p/ for some relevant ranges of p.

9.1.3 Properties of G.n; p/ In this subsection, we always let n  2 and G denote G.n; p/ and let ı WD ı.G.n; p//, m WD m.G.n; p//, and d WD d.G.n; p// D 2m=.n  1/. So d is well defined. For any vertex v, let dv denote the degree of v in G.

Typical Degrees The aim here is to show that m and d are a.a.s. concentrated around their expected values and that most of the vertices of G.n; p/ have degree close to d. Lemma 9.1.6 ([47]). For any function .n/    < 1, we have that the probability that jd  pnj   pn and jm  p n2 j  p n2 is at least 1  2 exp.A 2 pn2 / where A D 1=12. Lemma 9.1.7 ([47]). Let f  0 be any function of n such that f ! 1. Then, there exists a constant C > 0 such that for every f =n  p  1, the following holds in 1=3 G.n; p/ with probability at least 1  eC.pn/ . The number of vertices with degree not 1=3 in Œd  .pn/2=3 ; d C .pn/2=3  is at most n=eC.pn/ .

Maximum and Minimum Degree The following lemma bounds the probability that ı.G.n; p// deviates significantly from its expected value in the range 0:9 log n=.n  1/  p   log n=.n  1/, where   0:9 is a constant. This lemma will be applied when more precise probability bounds are required than those that would follow from the Chernoff’s inequalities in Theorem 9.0.1. Lemma 9.1.8 ([47]). Let   0:9 and 0 < < 1 be constants. Then there exists a constant C > 0 such that, for any functions p and ˛ of n satisfying 0 < ˛  1  and 0:9 log n=.n  1/  p   log n=.n  1/, the following holds:

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    e  1  log log n I .i/ Pr .ı  ˛p.n  1//  C exp log n  p.n  1/ 1  ˛ log ˛ 2 .ii/ Pr .ı > ˛p.n  1//      e  1 log n C exp p.n  1/ 1  ˛ log  log n C log log n : C n ˛ 2 The following lemma will be used when p is very close to the threshold function ˇ log n=.n  1/ described in Theorem 9.1.2, in order to have a fairly precise bound of the probability that ı.G.n; p// deviates slightly from pn=2. It is normally applied by choosing so that j jlog n is negligible compared to the other terms in (9.1) and (9.2). Lemma 9.1.9 ([47]). Let  > ˇ D 2=log.e=2/ and 0 < < 1=2 be constants. Then there exist positive constants C and D such that the following holds. For any functions p and of n satisfying j j  1  2 and 0:9 log n=.n  1/  p   log n=.n  1/, we have     .1 C / h 1 Pr ı  p.n  1/  C exp   log log n C Dj j log n ; and 2 ˇ 2 (9.1)      .1 C / log n h 1 Pr ı > p.n  1/  C C exp C log log n C Dj j log n ; 2 n ˇ 2 (9.2) where h D h.n/ is defined by p D

ˇ log nCh . n1

At this point, the reader may suspect that the relevant range of p for the study of the evolution of ı.G.n; p// corresponds to p D .log n=n/. Indeed, a careful application of Lemma 9.1.8 yields the following: if p c log n=n for some constant c > 1, then a.a.s. d c log n, and ı g.c/d, where g W Œ1; 1/ ! .0; 1/ is a strictly increasing continuous function with limc!1 g.c/ D 0 and limc!1 g.c/ D 0. The above claim will not be proved, as it is not needed in the argument, but rather several related statements are collected together in the following lemma. Lemma 9.1.10 ([47]). (i) For any p  0:9 log n=.n  1/, a.a.s. ı.G.n; p// D 0. (ii) For any constant > 0, there exist constants  > 1 and C > 0 such that, log n log n for every 0:9n1  p  n1 , we have that ı.G.n; p//  d.G.n; p// with C probability at least 1  n . (iii) Let  > 1 be a constant. There exist positive constants and C such that, for p   log n=.n  1/, we have that ı > p.n  1/ with probability at least 1  nC .

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(iv) For every constants 0 <  < 1 and C > 0, there exists a constant  > 0, such that for all p   log n=.n  1/, we have Pr.ı.G.n; p//   p.n  1//  nC . The last lemma gives the comparison of the minimum degrees of two random graphs G.n; p/ and G.n; pO /, when p and pO are close to one another. Lemma 9.1.11 ([47]). For any constants 1 < 1 < 2 and > 0, there exist positive constants and C such that, for any functions p and pO satisfying p// 1 log n=.n  1/  p  pO  2 log n=.n  1/ and pO =p  1  , ı.G.n;O  1  with ı.G.n;p// C probability at least 1  n .

Light Vertices Recall that an -light vertex was defined to be a vertex of degree at most ı C d. The following result shows that a.a.s. all -light vertices of G.n; p/ are at least three steps apart for a certain range of p. Lemma 9.1.12 ([47]). Suppose 0:9 log n=.n  1/  p   log n=.n  1/ for some constant   0:9. Then there exist constants > 0 and C > 0 such that the following holds in G.n; p/ with probability at least 1  nC . There is no pair of adjacent -light vertices, and no two -light vertices have a common neighbor. The following lemma is an extension of the above lemma in terms of two random graphs G1 G.n; p1 / and G2 G.n; p2 /, with 0  p1  p2 < 1, which are coupled together so that G1  G2 . This standard coupling can be achieved in the following way. Let G1 be distributed as G.n; p1 /, and let G2 the supergraph of G1 be obtained by adding each edge not in G1 independently with probability .p2  p1 /=.1  p1 /. Then G1  G2 and G2 has the same distribution as G.n; p2 /. Lemma 9.1.13 ([47]). Suppose 0:9 log n=.n  1/  p  p0   log n=.n  1/ for some constant   0:9. Let G1  G2 where G1 G.n; p/ and G2 G.n; p0 /. Then there exist constants > 0 and C > 0 such that the following holds in G.n; p/ and G.n; p0 / with probability at least 1  nC . Let S be the set of -light vertices in G1 . Then in G2 , there is no edge induced by S and no two vertices in S adjacent to a common vertex.

Graph Expansion For any sets S; S0  Œn, we recall that EG ŒS; S0  is the set of edges in G with one end in S and the other in S0 . Lemma 9.1.14 ([47]). Let f  0 be any function of n such that f ! 1 and  > 0 any fixed constant. Then, there exists a constant C > 0 such that for every f =n  2 p  1, the following holds in G.n; p/ with probability at least 1  eCpn . For every disjoint sets S; S0  Œn with jSj; jS0 j  n, we have jEG ŒS; S0 j  .d=4/jSjjS0 j=n.

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Lemma 9.1.15 ([47]). Let f  0 be any function of n such that f ! 1, and let ˛ > 0 be any fixed constant. Then, there exist constants  > 0 and C > 0 such that for every f =n  p  1, the following holds in G.n; p/ with probability at least 2 1  Ce.pn/ . For all s  n and every set S of size s, we have that jE.GŒS/j  ˛pns. Lemma 9.1.16 ([47]). Let  > 1 be a fixed constant. There exists a constant C > 0 such that for any p D p.n/   log n=.n  1/, the following holds in G.n; p/ with N  1:5ı, probability at least 1nC . For every S ¨ Œn with 2  jSj  n2, jEG ŒS; Sj where SN D Œn n S. Now Theorem 9.1.1 can be proved, as a consequence of Propositions 9.1.4 and 9.1.5. Proof of Theorem 9.1.1. For the rest of the argument, let ı WD ı.G.n; p// and let d WD d.G.n; p//. We split the argument into cases depending on the range of p. First observe that by Lemma 9.1.10 .i/ we can assume that p  0:9 log n=.n  1/, since for p  0:9 log n=.n  1/, the random graph G.n; p/ is a.a.s. disconnected and has minimum degree 0, so the statement of Theorem 9.1.1 holds trivially. Let 2 be a large enough constant so that for p  2 log n=.n  1/, we have ı > .3=4/d a.a.s. (see Lemma 9.1.10 (iv) and Lemma 9.1.6). Let < 1=2 be the constant given by Lemma 9.1.12 with  D 2 . Let 1 2 .1; 2 / be the constant given by Lemma 9.1.10 .ii/ with =4. For 0:9 log n=.n  1/  p  1 log n=.n  1/, we only need to show that G.n; p/ a.a.s. satisfies the hypothesis of Proposition 9.1.4. First, we note from Lemma 9.1.6 that d pn ! 1. Condition .a/ holds by our choice of 1 . Condition .b/ follows from Lemma 9.1.15 with any ˛ < =4, since d pn. Fix  as given by that lemma. Condition .c/ with D  2 =4 is a consequence of Lemma 9.1.14. Finally, we show that G.n; p/ a.a.s. satisfies the conditions in Proposition 9.1.5 for the range p  1 log n=.n  1/. First note that ı D ˝.d/ by Lemma 9.1.10 .iii/. (by our Condition .a0 / is satisfied for p  2 log n=.n  1/, since a.a.s. ı > .1C /d 2 choice of 2 and since < 1=2); and it is also satisfied for 1 log n=.n  1/  p  2 log n=.n  1/, since a.a.s. no -light vertices are adjacent nor have a common neighbors (by our choice of ). For condition .e0 /, note that t=d is bounded away from 0 since ı D ˝.d/. Therefore, the condition follows from Lemma 9.1.15 with ˛ D t=.8d/ (also using that a.a.s. d  pn=2), and this determines our choice of . Condition .b0 / holds a.a.s. by Lemma 9.1.7. Condition .c0 / holds a.a.s. by Lemma 9.1.14. Condition .d0 / holds a.a.s. by Lemma 9.1.16. t u Proof of Theorem 9.1.2. Thenumber of edges in G G.n; p/ is a binomial random  variable distributed as Bin . n2 ; p/. If p < 0:9 log n=n, then by Lemma 9.1.10 .i/, a.a.s. ı.G/ D 0, and thus a.a.s. ı.G/  d.G/=2. Assume p  0:9log n=n. By p Lemma 9.1.6, a.a.s. jd=2pn=2j  !n p, where d D d.G/. By Lemma 9.1.10 .iv/, there is a constant  > 0, such that for all p   log n=n, a.a.s. ı.G/  .3=4/pn. Hence, for p in this range, a.a.s. ı.G/ > d=2. Now we only consider 0:9 log n=n  p   log n=n.

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Let !n be a positive-valued function of n that goes to infinity arbitrarily slowly as p nCf n ! 1, and let D !n = pn. Define f D f .n/ by p D ˇ log . To prove statement n1 .ii/, we assume f  ˇ log log n=2 C !n . By Lemma 9.1.9 (with h D f ), we have      f 1 1 D o.1/; Pr ı  .1 C /pn D O exp   log log n C O. log n/ 2 ˇ 2 p as f =ˇ  log log n=2  !n , whereas log n D O.!n p/ D o.1/. Moreover, by Lemma 9.1.6, Pr.d=2  .1 C /pn=2/ D o.1/. Thus, a.a.s. ı > d=2 and so n .G/ D bd=2c by Theorem 9.1.1. This completes the proof of statement .ii/. On the other hand, if f  ˇ log log n=2  !n , then f =ˇ C log log n=2  !n , and thus by Lemma 9.1.9 (with h D f and replaced by  ), 

    log n f 1 1 Pr ı > .1 C /pn D O C exp C log log n C O. log n/ D o.1/: 2 n ˇ 2

Again by Lemma 9.1.6, Pr.d=2  .1  /pn=2/ D o.1/. Thus, a.a.s. ı < d=2, and thus n .G/ D ı.G/ by Theorem 9.1.1, which completes the proof of statement .i/. t u Now we are in a position to give the proof of Theorem 9.1.3. A standard tool to investigate the random graph process G0 ; : : : ; Gm ; : : : ; G.n/ is the related continuous 2 random graph process .Gp /p2Œ0;1 defined as follows. For each edge e of the complete graph with vertex set Œn, we associate a random variable Pe uniformly distributed in Œ0; 1 and independent from all others. Then, for any p 2 Œ0; 1, we define Gp to be the graph with vertex set Œn and precisely those edges e such that p  Pe . Note that for each p, Gp is distributed as G.n; p/. This provides us with a useful way of coupling together G.n; p/ for several values of p, since p  p0 implies Gp  Gp0 . Moreover, let p.m/ D minfp 2 Œ0; 1 W Gp has at least m edgesg. Then, Gp.0/ ; : : : ; Gp.m/ ; : : : ; Gp..n// 2 is distributed as G0 ; : : : ; Gm ; : : : ; G.n/ , since all Pe are different with probability 1. 2 For more details on the connection between .Gp /p2Œ0;1 and .Gm /0m.n/ and further 2 properties, we refer the reader to [63]. In [47], they proved several statements that hold a.a.s. simultaneously for all m in the random graph process .Gm /0m.n/ . To do so, it is often convenient to use small 2 bits of the continuous random graph process as follows. Given p0 and p1 as functions of n such that 0  p0  p1  1, we consider .Gp /p0 pp1 . Let m0 D m.Gp0 / and m1 D m.Gp1 /. (Note that m0 and m1 are random variables with m0  m1 , since Gp0  Gp1 .) Color all edges of Gp0 D Gm0 red and the remaining m1  m0 edges in Gp1 n Gp0 blue. Then, Gm0 ; Gm0 C1 ; : : : ; Gm1 can be interpreted as a random graph process in which one sequentially adds blue edges to Gm0 , so that each Gm has the m0 red edges of Gm0 together with the first mm0 blue edges one adds in the process. This interpretation will be used many times throughout the argument. The following result is stronger than part .ii/ of Theorem 9.1.3 and is also used in the argument for part .i/.

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Theorem 9.1.17 ([47]). Consider the random graph process .Gp /0p1 . We have that a.a.s.: log n=2/!.1/ (i) For all p  ˇ.log nlogn1 , we have ı.Gp /  d.Gp /=2; and ˇ.log nlog log n=2/!.1/ , we have ı.Gp / > d.Gp /=2. Moreover, for (ii) For all p  n1 every constant 0 <  < 1, there is a constant  > 0 such that a.a.s.: (iii) For all  log n=.n  1/  p  1, we have ı.Gp / > d.Gp /.

Proof. First, we prove statement .iii/. We will show that for every 0 <  < 1, there exists  > 0 such that a.a.s. ı.Gm / > d.Gm /; for all m  m0 D .=4/n log n

(9.3)

Then, let p0 D  log n=.n  1/. By Chernoff’s bounds in Theorem 9.0.1, a.a.s. m.G.n; p0 // > m0 , i.e., a.a.s. p.m0 / < p0 . It follows then that  a.a.s. ı.Gp / > d.Gp / for all p  p0 . Now we prove (9.3). For each m, let pN D m= n2 . Then Pr .ı.Gm /    2m=.n  1// D Pr .ı.G.n; pN //   pN n j m.G.n; pN // D m// : By the choice of pN , h.i/ D Pr .m.G.n; pN // D i/ is maximized at i D m. Hence, Pr .m.G.n; pN // D m/  n2 . Thus, Pr.ı.Gm /    2m=.n  1// 

Pr.ı.G.n; pN //   pN n/  n2 Pr.ı.G.n; pN //   pN n/: Pr.m.G.n; pN // D m/

By Lemma 9.1.10 (iv), for every 0 <  < 1, we can choose  > 0 sufficiently large such that the probability on the right-hand side above is less than 1=n5 for every m  .=4/n log n (correspondingly, pN  .=2/log n=.n  1//. Hence, taking a union bound over the O.n2 / possible values of m, we deduce that claim (9.3) is true with probability at least 1  O.n1 /. p Next, we prove statements .i/ and .ii/. Let f D o. log n/ be a function that goes to 1 arbitrarily slowly, as n ! 1. Let pi D .ˇ.log nlog log n=2/f 2 if /=.n1/, and let qi D .ˇ.log n  log log n=2/ C f 2 C if /=.n  1/, for each i  1. Let T be the smallest integer such that pT  0:9 log n=.n  1/, and redefine pT D 0:9 log n= .n  1/. Let  be the constant satisfying statement .iii/ with  D 3=4. Let T 0 be the smallest integer such that qT 0   log n=.n  1/, and redefine qT 0 D  log n=.n  1/. Obviously, T; T 0 D O.log n/. Claim 1. There exists a positive constant C, such that, for every 1  i < T, Pr.ı.Gpi / > d.GpiC1 /=2/  C.f i C log n=n/; and for every 1  i < T 0 , Pr.ı.Gqi /  d.GqiC1 /=2/  C.f i C n1 /:

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Proof of Claim 1. By Lemma 9.1.10 .i/ and from the monotonicity of ı.Gp / with respect to p, a.a.s. for all p < pT D 0:9 log n.n  1/, we have ı.Gp / D 0, and thus ı.Gp /  d.Gp /=2 holds. By Claim 1, with probability at least X C.f i C log n=n/ D 1  o.1/; 1 1i d.Gp /=2 for all p satisfying q1  p  qT 0 D  log n=.n  1/. Similarly as in the previous argument, with probability at least

1

X

C.f i C n1 / D 1  o.1/;

1i d.Gp /=2 for all p  statement .ii/.

ˇ.log nlog log n=2/C!.1/ , n1

as required in

Finally, we prove Claim 1. In this argument the asymptotic statements are uniform for all p 2 ŒpT ; p1  [ Œq1 ; qT 0 . By Lemma 9.1.6, for  D n1=3 and a positive constant A, Pr.jd.Gp /  pnj >  pn/  exp.A 2 n2 p/ D o.n1 /:

(9.4)

i piC1 n may be written as ı.Gpi / > 1C pi .n  1/ for Note that the event ı.Gpi / > 1 2 2 some negative i D  .f = log n/. Hence, using (9.4) and also Lemma 9.1.9 with

D i and h.n/ D ˇ log log n=2  f 2  if , we get that, for every 1  i < T,

! d.GpiC1 / Pr ı.Gpi / > 2     1 piC1 n C Pr d.GpiC1 / < .1   /piC1 n  Pr ı.Gpi / > 2  2   f  if log n C exp C O.f / C o.n1 / DO n ˇ   log n : D O f i C n

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Similarly, we write ı.Gqi /  1C qiC1 n as ı.Gqi /  2 i qi .n  1/ for some i0 D 2 .f = log n/. Using (9.4) and Lemma 9.1.9 with D i0 and h.n/ D ˇ log log n=2C f 2 C if , we obtain, for every 1  i < T 0 , ! d.GqiC1 / Pr ı.Gqi /  2     1C qiC1 n C Pr d.GqiC1 / > .1 C  /qiC1 n  Pr ı.Gqi /  2    2 f  if C O.f / C o.n1 / D O exp ˇ   D O f i C 1=n : t u log n Proof of Theorem 9.1.3. We first prove statement .ii/. Let p1 D .1 =2/ˇ and let n1 .1 =2/ˇ log n . For i D 1; 2, the number of edges in G.n; pi / follows the binomial p2 D n1  distribution Bin . n2 ; pi /. By Lemma 9.1.6, we have that a.a.s. m.G.n; p1 //     1 1 p n D .1 /ˇn log n=2 and m.G.n; p2 //  1 =2 p2 n2 D .1C /ˇn log n=2. 1 =2 1 2 Then, Theorem 9.1.3 .ii/ follows immediately from Theorem 9.1.17.

We now proceed to prove statement .i/ of Theorem 9.1.3. Recall that for any graph G, t.G/ D minfı.G/; d.G/=2g. First, define p0 D 0:9 log n=.n  1/, p1 D 1 log n=.n  1/, and p2 D 2 log n=.n  1/, for some constants 1 < 1 < 2 that we specify later. We prove the statement separately for .Gp /p0 pp1 , .Gp /p1 pp2 , and .Gp /p2 p1 . For .Gp /0pp0 it is trivially true since a.a.s. ı.Gp / D 0 for all 0  p  p0 , by Lemma 9.1.10 (i) and the monotonicity of ı.Gp / with respect to p. Part 1 (p0  p  p1 ) Let > 0 be a constant chosen to satisfy Lemma 9.1.13 with  D 1:1. Pick a sufficiently small constant 1 < 1 < 1:1, and recall p0 D 0:9 log n=.n  1/ and p1 D 1 log n=.n  1/. From Lemma 9.1.6, a.a.s. d.Gp1 /  .4=3/d.Gp0 /:

(9.5)

Moreover, in view of Lemma 9.1.10 (ii), we assume that 1 is small enough so that a.a.s. ı.Gp1 /  . =16/d.Gp1 /  . =12/d.Gp0 /:

(9.6)

Color edges in Gp1 so that all edges in Gp0 are colored red and all edges in Gp1 nGp0 are colored blue. For each vertex v 2 Gp1 , the red (blue) degree of v is the number of red (blue) edges incident with v. Let S be the set of -light vertices of Gp0 . Since ı.Gp0 / D 0 a.a.s., the -light vertices are a.a.s. precisely those vertices with degree at most d.Gp0 / in Gp0 . By the choice of and Lemma 9.1.13, a.a.s. the vertices in S induce no edges and have no common neighbors in the whole process .Gp /p0 pp1 as blue edges are added.

9.1 Results for the Spanning-Tree Packing Number

125

For each p0  p  p1 , let Sp be the set of .11 =16/-light vertices of Gp (i.e., vertices of degree at most ı.Gp / C .11=16/ d.Gp / in Gp ). Note that a.a.s. S contains Sp for all p in this range, since for any v 2 Sp , its degree in Gp0 (i.e., the red degree of v in Gp ) is at most ı.Gp / C .11=16/ d.Gp /  ı.Gp1 / C .11=16/ d.Gp1 /  . =12/d.Gp0 / C .11=12/ d.Gp0 / D d.Gp0 / where we used (9.5) and (9.6). We just showed that a.a.s. in .Gp /p0 pp1 the set of .11 =16/-light vertices of Gp induce no edges and have no common neighbors. Moreover, from (9.6) and by monotonicity of ı.Gp / and d.Gp / with respect to p, we have that a.a.s. ı.Gp /  ı.Gp1 /  . =12/d.Gp0 /  . =12/d.Gp / in the whole process .Gp /p0 pp1 . Putting all that together, we have that a.a.s. the conditions of Proposition 9.1.4 are satisfied in .Gp /p0 pp1 (replacing by .11=16/ ), and therefore a.a.s. n .Gp / D ı.Gp / simultaneously for all p in this range. Part 2 (p1  p  p2 ) Recall that p1 D 1 log n=.n  1/ and p2 D 2 log n= .n  1/, where 1 is as in Part 1 and 2 > 1 is a sufficiently large constant. In view of Theorem 9.1.17 .iii/, we assume that 2 is large enough so that a.a.s. ı.Gp /  .3=4/d.Gp / in the whole process .Gp /p2 p1 . Define qi D .1 C 1= log n/i p1 for each i D 0; 1; 2; : : :, and let T be the smallest integer such that qT  p2 . Redefine qT D p2 . We have T  2 log.2 =1 / log n D O.log n/, since eventually .1 C 1= log n/2 log.2 =1 / log n > 2 =1 : To prove the statement for .Gp /p1 pp2 , it suffices to see that for every 0  i  T 1, we have n .Gp / D bt.Gp /c throughout the process .Gp /qi pqiC1 with probability at least 1  1= log2 n and then simply take a union bound over all i. Let be as in Lemma 9.1.13 (putting  D 2 ), and fix 0  i  T  1. We verify that with probability at least 1  1= log2 n, all conditions (a’)–(e’) of Proposition 9.1.5 are satisfied in .Gp /qi pqiC1 . We color as before the edges of Gqi red and the additional edges in GqiC1 n Gqi blue. Let S be the set of vertices that are -light in Gqi (they have red degree at most ı.Gqi / C d.Gqi //. For each qi  p  qiC1 , define Sp to be the set of vertices that are

=2-light in Gp . From Lemma 9.1.6 and Lemma 9.1.11, we have that d.Gqi / d.GqiC1 / and ı.GqiC1 /  .1 C =3/ı.Gqi /

(9.7)

with probability at least 1  nC , for some small enough constant C > 0 not depending on i. These equations imply that S  Sp for all p in our range, since the red degree of any vertex in Sp is at most

126

9 Generalized Connectivity for Random Graphs

ı.Gp / C . =2/d.Gp /  ı.GqiC1 / C . =2/ d.GqiC1 /  .1 C =3/ı.Gqi / C . =2  o.1//d.Gqi /  ı.Gqi / C d.Gqi / where we also used the trivial fact that ı.Gqi /  d.Gqi /. By Lemma 9.1.13, with probability at least 1nC , the vertices in S do not get common neighbors or induced edges as the blue edges are added in .Gp /qi pqiC1 . This implies condition .a0 / replacing by =2. By Lemma 9.1.10 .iii/, there exists a constant  > 0 such that, uniformly for all p 2 Œp1 ; p2 , we have that ı.Gp /   pn with probability at least 1  nC . Therefore, t.Gqi /   0 qiC1 n with probability at least 1  nC for a positive constant  0 not depending on i. By applying Lemma 9.1.15 to GqiC1 with ˛ <  0 =4, we deduce that condition .e0 / holds with probability at least 1  nC during all the process .Gp /qi pqiC1 . This determines our choice of . Next, observe that Gqi satisfies condition .b0 / with probability at least 1eC.log n/1=3 , by Lemma 9.1.7, and also condition .c0 / with probability at least 1  nC , by Lemma 9.1.14. Therefore, in view of (9.7), both conditions .b0 / and .c0 / hold simultaneously in all .Gp /qi pqiC1 with probability at least 1  eC.log n/1=3 . Condition .d0 / holds trivially for sets S of size 1. For larger sets, Lemma 9.1.16 applied to Gqi together with (9.7) implies that this condition holds in all .Gp /qi pqiC1 with probability 1  nC . Taking the union bound for all 0  i  T  1 D O.log n/, we have that a.a.s. n .Gp / D bt.Gp /c throughout the process .Gp /p1 pp2 by Proposition 9.1.5. Part 3 (p2  p  1) Let 2 be as in part 2 and p2 D 2 log n=.n  1/. Recall from the definition of 2 that a.a.s. ı.Gp /  .3=4/d.Gp / in the whole process .Gp /p2 p1 , and therefore condition .a0 / in Proposition 9.1.5 holds. Define qi D .1C1= log n/i p2 for each i D 0; 1; 2; : : :, and let T be the smallest integer such that qT  1. Redefine 2 qT D 1. Observe that T  3 log2 n, since eventually .1 C 1= log n/3 log n  n2:5 . The same argument as in part 2 shows that for every 0  i  T  1, conditions .b0 /.e0 / in Proposition 9.1.5 are satisfied throughout the process .Gp /qi pqiC1 with probability at least 1  1= log3 n. Taking the union bound over all i, we conclude that a.a.s. all conditions in Proposition 9.1.5 hold and therefore n .Gp / D bd.Gp /=2c, during the whole process .Gp /p2 p1 . t u

9.2 Results for the Generalized 3-Connectivity For the vertex connectivity, Bollobás and Thomason [18] gave the following result. Theorem 9.2.1 ([18]). If ` 2 N and y 2 R are fixed and M D ` log log n C y C o.1// 2 N, then y=`Š

Pr . .G .n; M// D `/ ! 1  ee

n .log n 2

y=`Š

and Pr . .G .n; M// D ` C 1/ ! ee

C

:

9.2 Results for the Generalized 3-Connectivity

127

For a general k, it is difficult to get a similar result as above for the generalized k-connectivity. So Gu, Li, and Shi [53] focused their attention on the generalized 3-connectivity of random graphs and got the following theorem, which could be seen as a generalization of Theorem 9.2.1. Theorem 9.2.2 ([53]). Let `  1 be a fixed integer. Then pD

log n C .` C 1/ log log n  log log log n n

is a sharp threshold function for the property 3 .G.n; p//  `. The sketch of their proof for Theorem 9.2.2 is as follows. At first, they proved that there exists a constant c such that if p0 < c log nC.`C1/ log nlog nlog log log n , then 3 .G.n; p// < ` almost surely holds. Then, they showed that for any three vertices in G.n; p/, where p D log nC.`C1/ log nlog nlog log log n , there almost surely exist three trees of some typical depths rooted at these three vertices, respectively. By combining some branches of these trees, they constructed ` internally disjoint Steiner trees connecting any three vertices, and hence 3 .G.n; p//  `. The following lemma is needed. A property Q is called convex if F  G  H and F satisfies Q; then H that satisfies Q implies that G satisfies Q, where F; G; H are some graphs. Set N D 12 n.n  1/. Lemma 9.2.3 ([17]). If Q is a convex property and p.1  p/N ! 1, then G.n; p/ almost surely satisfies Q if and only if for every fixed x, G.n; M/ almost surely satisfies Q, where M D bpN C x.p.1  p/N/1=2 c. To establish a sharp threshold function for a graph property, the proof should be twofold. We first show one easy direction. Proposition 9.2.4 ([53]). 3 .G.n; 12 log nC.`C1/ log nlog nlog log log n //  `  1 almost surely holds. Proof. Let p D log nC.`C1/ log nlog nlog log log n and M 0 D b 12 pN C xf 12 p.1  12 p/Ng1=2 c for any x 2 R, i.e., M 0 D n4 .log n C .` C 1/ log log n  log log log n C o.1//. It is easy to check that 12 p.1  12 p/N ! 1. Let M1 D n2 .log n C .`  1/ log log n C y C o .1// 2 N. By Theorem 9.2.1, we have y=.`1/Š

Pr . .G .n; M1 // D `  1/ ! 1  ee

:

Hence, for any " > 0, there exist an N 0 2 N and a Y 2 RC , such that for any y < Y, y=.`1/Š

1  ee

 Pr . .G .n; M1 // D `  1/
N1 . Then      Pr  G n; M 0  `  1 D

1 X

     Pr  G n; M 0  `  1 j  .G .n; M1 // D i  Pr . .G .n; M1 // D i/

iD0

      Pr  G n; M 0  `  1 j  .G .n; M1 // D `  1  Pr . .G .n; M1 // D `  1/      D Pr  G n; M 0  `  1;  .G .n; M1 // D `  1 D Pr . .G .n; M1 // D `  1/ Hence, for any n > maxfN 0 ; N1 g, we have      " y=.`1/Š 1  Pr  G n; M 0  `  1  1  Pr . .G .n; M1 // D `  1/ < ee C < ": 2

Thus,  .G .n; M 0 //  `  1 almost surely holds. Obviously, the property that the connectivity of a given graph is at most `1 is a convex property. By Theorem 4.1.7 and Lemma 9.2.3, 3 .G.n; 12 log nC.`C1/ log nlog nlog log log n //  `  1 almost surely holds. t u We leave with the other direction stated below. Proposition 9.2.5 ([53]). 3 .G.n; log nC.`C1/ log nlog nlog log log n //  ` almost surely holds. From now on, p is always log nC.`C1/ log nlog nlog log log n . Similarly to the proof of Proposition 9.2.4, we can prove the following result. Lemma 9.2.6 ([53]). `  .G.n; p//  ` C 1 almost surely holds. From Lemma 9.2.6, we know that the minimum degree of G.n; p/ is a.s. at least `. Let G D G.n; p/ and D D logloglogn n . Almost surely, the diameter of G is asymptotically equal to D; see, for example, [17]. We call a vertex v large if its degree dG .v/ is at n least log and small otherwise. 100 Lemma 9.2.7 ([53]). Almost surely, there do not exist two small vertices within distance at most 34 D in G. Lemma 9.2.8 ([53]). For a fixed t 2 N and 0 < ˛ < 1, almost surely, there does not exist a subset S  V.G/ such that jSj  ˛tD and e.GŒS/  jSj C t. Remark 9.2.9 ([53]). Let T be a rooted tree of depth at most 3D=4, and let v be a vertex not in T, but with b neighbors in T. Let S consist of v, the neighbors of v in T, and the ancestors of these neighbors. Then jSj  3bD=4 C 1 C 1  4bD=5 and e.GŒS/ D jSj C b  2. It follows from the proof of Lemma 9.2.8 with ˛ D 4=5 and t D 16 that we must have b  18, with probability 1  o.n.1=5o.1//t /  1  o.n3 /.

9.2 Results for the Generalized 3-Connectivity

129

Remark 9.2.10 ([53]). Let B be a set of at most ` vertex-disjoint paths and trees, each containing at most 5D=2 edges, and let v be a vertex not in B, but with c neighbors in B. Let S D fvg [ V.B/, jSj  5`D=2 C ` C 1  11`D=4, and e.GŒS/ D jSj C c  `  1. By Lemma 9.2.8 with ˛ D 1=4 and t D 11`, we deduce that with probability at least 1  o.n3 /, c  12` C 1. We first deal with large vertices. The following lemma points out that for every pair of large vertices in V.G/, there exists a special subgraph containing them, which can be used to find Steiner trees connecting given vertices. Recall that a t-ary tree with a designated root is a tree whose non-leaf vertices all have exactly t children. A tree search in which the adjacency lists of the vertices of a tree T are considered on a first-come first-served basis, that is, in increasing order of their time of incorporation into T, is known as breadth-first search (simply, we write BFS). For more details on BFS, we refer to [20, Chapter 6]. Lemma 9.2.11 ([53]). Let " D ".n/ D log 1log n . Then, almost surely, for any pair of large vertices u and v 2 V.G/, there exists a subgraph Gu;v of G that consists n of two vertex-disjoint log -ary trees Tu and Tv rooted at u and v, respectively, such 101 that each of them has depth . 34  "/D. Proof. For any pair of large vertices u and v, we will show that the two trees described in Lemma 9.2.11 exist with probability 1  o.n3 /. We first grow a tree from u using BFS until it reaches depth . 34  "/D. Then we grow a tree starting from v again using BFS until it reaches depth . 34  "/D. We use the notation Si .x/ for the number of vertices at depth i of the BFS tree rooted at x. As growing Tu , when we grow the tree from a vertex x at depth i to depth i C 1, there may exist some bad edges which connect x to the vertices already in Tu . From Remark 9.2.9, there exist at most 18 bad edges from x with probability 1  o.n3 /. For small vertices, from Lemma 9.2.7, in the first 3D=8 levels, there exists at most one small vertex at each level a.s. Furthermore, once a small vertex appears in the BFS tree, there will be no small vertex in the subtree rooted at that small vertex. Though there may be more than one small vertex in depth 3D=8 C 1, the number of them will not exceed the number of branches at root u, since one branch contains at most one small vertex in depth 3D=8C1, a.s. Then in depth 3D=8C2, the number of small vertices of that level will be no more than the number of vertices in the depth 3D=8 C 1 contained in the branches which have no small vertex in the previous levels. For the remaining levels of that BFS tree, we can conclude the similar result. Note that there will exist no small vertex in the following levels of branches which contain small vertices in depth at least 3D=8 C 1 of that BFS tree, a.s. Hence the number of small vertices contained in each level is much smaller than the increase of the number of vertices in each level. Denote by ti .u/ the number of small vertices of depth i. Thus we get the following recursion: SiC1

.u/

 

   log n .u/ log n  18 Si .u/  ti .u/  Si 100 101

130

9 Generalized Connectivity for Random Graphs

We call the operation of deleting some vertices from a tree as prune a tree. Clearly, n we can make the current BFS tree a log -ary tree by pruning. 101 Similarly, we grow Tv . The only difference is that now we also say that an edge is bad if the other endpoint is in Tu . Hence, SiC1 .v/ 



  log n .v/  log n  36 Si .v/  ti .v/  Si : 100 101 t u

After pruning, we can obtain the required subgraph Gu;v . We are now in the position to give a proof of Proposition 9.2.5.

Proof of Proposition 9.2.5. It suffices to show that for any three vertices, we can find at least ` internally disjoint Steiner trees connecting them in G. Given three vertices u, v, and w, we first assume that they are all large vertices. By n Lemma 9.2.11, we construct two vertex-disjoint log -ary trees Tu and Tv rooted 101 at u and v, respectively, such that each of them has depth . 43  "/D. For every tree T, denote by L.T/ the set of leaves of T. Let u1 ; : : : ; u log n .resp: v1 ; 101 : : : ; v log n / be the vertices in the first depth of Tu .resp: Tv /. For each ui .resp: vi /, 101 denote by Tui .Tvi / the subtree of Tu .resp: Tv / of depth .3=4  "/ D  1 rooted at n . We call these Tui .resp: Tvi / vice trees. ui .resp: vi /, i D 1; : : : ; log 101 For a fixed Tui , let the random variable Ai denote the number of edges between L.T L.Tv /. Then Ai follows the binomial distribution, i.e.,  ui / and .3=4"/D  .3=4"/D1  log n log n Ai Bin  101 ; p . The expectation value of Ai is 101     9:2.3=4"/ 1 log n 2.3=4"/D1 101 log n 2.3=4"/D E.Ai / D p   101n 2 2" log log n : 101 n 101 By Chernoff’s bounds (see Theorem 9.0.1),   1 2" 9:2.3=4"/ 1 1 1 1 100 2 log log n / E .Ai /  e 2  1012 E.Ai  e 2  1012 101n Pr Ai < 101 1 2" 9:2.3=4"/ 1 1 1 2 log log n 2  101 log n n

 log n

De 1

1

1

D n 2  101 log n n 1 o.1/

 nn 2

!

1 2" 9:2.3=4"/ 2 log log n

:

Now, for a fixed Tvj , let Aij denote the number of edges between L.Tui / and L.Tvj /.  .3=4"/D1  .3=4"/D1  log n log n Then Aij Bin  ; p , and hence 101 101

9.2 Results for the Generalized 3-Connectivity

 E.Aij / D

D

log n 101

131

2.3=4"/D2 p

  1012 log n 2.3=4"/D 2 log n n log2 n 101

/ 1012  2 12 2" 9:2log.3=4" log n : n log n

Also, we can deduce that E.Aij /  we have

1012 log n

1

 n 2 2"

9:2.3=4"/ log log n

1012 2

. By Chernoff’s bounds,

1 2" 9:2.3=4"/ 2 log log n

   e7E.Aij / e7 log n n Pr Aij > 8E Aij  8E.A /  9:2.3=4"/ 1 8 ij 1012 2 2" log log n 88 log n n e

D e

9:2.3=4"/ 1 1012 14 2 2" log log n log n n

9:2.3=4"/ 1 1012 8log 8 2 2" log log n n log n 2

1 2" 9:2.3=4"/ log log n

 2:64101 n 2 2

Dn

log n

1 o.1/

 nn 2

:

1 o.1/

2=5

n By the union bounds, with probability at least 1 log o.nn 2 /  1o.nn /, 101 we have that for every Tvj , the number of edges between L.Tui / and L.Tvj / is at most 2.3=4"/D2  1 o.1/   n 8E Aij =8p log . Therefore, with probability at least 1  o.nn 2 / 101 n2=5

o.n

n2=5

/ D 1  o.n

/, there are at least

 log n 2.3=4"/D1 p 101   log n 2.3=4"/D2

100 101

8



101

p

D

100 1012 8

log n

vice trees Tvj such that vertices in L.Tui / and L.Tvj / can be connected with edges. 2=5 n Moreover, using the union bounds, with probability at least 1  log o.nn / 101 1=5  1  o.nn /, each vice tree of Tu can be connected to 101100 2 8 log n vice trees of Tv with edges. Hence there are at least 101100 log n pairs fT ; T u vj g such that vertices 2 8 i of L.Tui / and L.Tvj / can be connected by edges. For convenience, let a log n D 101100 2 8 log n. Without loss of generality, assume these a log n pairs be Tur and Tvr , r D 1; 2;    ; a log n. Now we show that, for the remaining large vertex w, we can find at least ` internally disjoint Steiner trees connecting fu; v; wg. Note that we can assume that w is not in Tu and Tv , since otherwise we can prune the tree by deleting the subtree rooted at w (just like the n way to deal with small vertices) and we can still obtain log -ary trees rooted at u 101 and v, respectively. With the similar argument in the proof of Lemma 9.2.11, we can construct a log n -ary tree Tw of depth .1=4 C 2"/D rooted at w such that Tu , Tv , Tw are vertex 101 disjoint. Note that at this time the number of small vertices in each level is at most one. Let w1 ; : : : ; w log n be the vertices of the first depth of Tw . Let Qi denote the 101

132

9 Generalized Connectivity for Random Graphs

tree induced by the edges in E.Tui / [ E.Tvi /, where i D 1; 2; : : : ; a log n. Then .3=4"/D1  n jQi j > 2 log . For any fixed Qj , let qj denote the probability that there 101 exists at least one edge between Twj and Qj . Then qj D 1  Pr .there is no edge between Twj and Qj / D 1  .1  p/jQj jjTwj j     log n .3=4"/D1 log n .1=4C2"/D1 2 101  101

 1  .1  p/ 2

 1e

log n n



log n 101

.1C"/D2

2

4:6.1C"/ " log log n

2 log n 1012 n

D 1e

log n

1012

4:6.1C"/ " log log n

D 1  e2 log n n

:

For any i ¤ j, we have qi D qj . Let q D qi D qj , and let A be the event that there are at most `  1 pairs of fTwr ; Qr g, such that there exist edges between Twr and Qr , where r D 1; : : : ; a log n. By the upper bound of the probability that A happens, we can deduce that Pr .A / 

 `1  X a log n i q .1  q/a log ni i iD0

  a a log n  e `1 .1  q/ 2 log n ` `1   2 " 4:6.1C"/ a a log n  e `1 2 101 log log n  log n 2