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CHROMATICITY and ADJOINT POLYNOMIALS of GRAPHS °C H CHROMATICITY AND ADJOINT POLYNOMIALS OF GRAPHS °c H. Zhao, Enschede, 2005 No part of this work may be reproduced by print, photography or any other means without the permission in writing from the author. Printed by W¨ohrmannPrint Service, The Netherlands. ISBN: 90-3652176-9 CHROMATICITY AND ADJOINT POLYNOMIALS OF GRAPHS PROEFSCHRIFT ter verkrijging van de graad van doctor aan de Universiteit Twente, op gezag van de rector magnificus, prof. dr. W.H.M. Zijm, volgens besluit van het College voor Promoties in het openbaar te verdedigen op donderdag 16 juni 2005 om 13.15 uur door Haixing Zhao geboren op 14 mei 1969 te Qinghai, China Dit proefschrift is goedgekeurd door de promotoren prof. dr. Cornelis Hoede en prof. dr. Xueliang Li Acknowledgements The research presented in this thesis could not have been completed without the help and support of many persons. I would like to express my sincere gratefulness to them. Here I would like to mention some of them in particular. First of all I am very grateful to my promoters Prof. Dr. Kees Hoede and Prof. Dr. Xueliang Li for their scientific guidance, for their valuable discussions and comments, and for their inspiration and encouragement. Past years they gave me the opportunity to study at the University of Twente and Prof.Dr.Li invited me several times to study at the Northwestern Polytechnical University in Xi’an and at the Nankai University in Tianjin. Their stimulating enthusiasm and optimism created an excellent working atmosphere. It is a pleasure to work under their supervision. I would like to thank Prof. Dr. Ir. Broersma, Prof. Dr. A.E. Brouwer, Prof. Dr. Ir. R. Martini and Prof. Dr. G.J. Woeginger for their willingness to partitcipate in my graduation committee. I am also thankful to the members of the group of Discrete Mathematics and Mathematical Programming at the University of Twente for providing a pleasant working environment. Special thanks go to Dr. G. Still and Dini for giving me help and support. I wish to convey my thanks to Prof. Ruying Liu, Prof. Mingkui Zhu and Prof. Chengfu Ye for their help and support throughout my study. Thanks also go to Prof. Dr. Xiaodong Liu, Prof. Dr. Shenggui Zhang and Prof. Ruihu Li for giving me help during my study in Xi’an. I also thank the other Ph.D.students, namely, T. Br¨uggemann,P.S. Bonsma, G. Bouza, A.F. Bumb and A.N.M. i Salman, Ligong Wang, Xinhui Wang and Jichang Wu for making my time of Ph.D studies so enjoyable. Finally, I want to thank my parents, brothers and sisters. They provided me generous support and encouragement in these years, even if they did not always understand what I was doing. Last, but not least, I gratefully acknowl- edge my wife Zhanlan Li and my daughter Longjiao Zhao for their support and love. Haixing Zhao May 2005, Enschede Preface This thesis is the result of almost three years of research in the field of chro- maticity of graphs between September 2002 and March 2005. After an intro- ductory chapter the reader will find five chapters that contain two main parts within this research field. The two parts have strong connections with each other. The first part, Chapters 2 and 3, is on algebraic properties and roots of adjoint polynomials. The second part, Chapters 4, 5 and 6, studies mainly the chromaticity of some classes of graphs, that is dense graphs, complete mul- tipartite graphs and general multipartite graphs. Some results of this thesis have been published in journals: see the following list. 1. On the minimum real roots of the σ-polynomials and chromatic unique- ness of graphs, Discrete Mathematics 281 (2004), 277-294 (with X. Li, R. Liu and S. Zhang). 2. A complete solution to a conjecture on chromatic uniqueness of complete tripartite graphs, Discrete Mathematics 289 (2004), 175-179 (with R. Liu and C. Ye). 3. On problems and conjectures on adjointly equivalent graphs, Discrete Mathematics 295 (2005), 203-212 (with X. Li and R. Liu). 4. The chromaticity of certain complete multipartite graphs, Graphs and Combinatorics 20 (2004), 423-434 (with X. Li, R. Liu and C. Ye). 5. On properties of adjoint polynomials of graphs and their applications, Australasian Journal of Combinatorics 30 (2004), 291-307 (with X. Li, iii R. Liu and L. Wang). 6. A note on adjoint polynomials and uniquely colorable graphs, Journal of Combinatorial Mathematics and Combinatorial Computing 45(2003), 123-128 (with R. Liu). Contents Acknowledgements i Preface iii 1 Introduction 1 1.1 Basic definitions and knowledge . 2 1.2 The adjoint polynomials of graphs . 6 1.3 The chromaticity of some dense graphs . 13 1.4 The chromaticity of multipartite graphs . 19 2 Some Properties of Adjoint Polynomials of Graphs 29 2.1 Introduction . 29 2.2 The divisibility of adjoint polynomials of some graphs . 31 2.3 Adjoint polynomials of graphs and uniquely colorable graphs . 37 3 On the Roots of Adjoint Polynomials of Graphs 41 3.1 Introduction . 41 3.2 Some basic properties of the minimum real roots of an adjoint polynomial . 42 3.3 Graphs G with ¯(G) 2 [¡4; 0] . 51 p 3.4 Graphs G with ¯(G) 2 [¡(2 + 5); ¡4) . 54 3.5 The complex roots of adjoint polynomials of graphs . 59 4 The Chromaticity of Some Dense Graphs 65 4.1 Introduction . 65 4.2 Some basic results . 66 v 4.3 The chromaticity of graphs G with ±(G) ¸ jV (G)j ¡ 3 . 67 4.4 A solution to two conjectures on adjointly equivalent graphs . 79 4.5 The adjoint uniqueness of the union of T -shaped trees . 85 4.6 An invariant R2(G) for adjointly equivalent graphs and its ap- plication . 90 4.7 A new invariant R3(G) for adjointly equivalent graphs and its application . 99 5 The Chromaticity of Complete Multipartite Graphs 105 5.1 Introduction . 105 5.2 Some basic lemmas . 107 5.3 The chromaticity of complete tripartite graphs . 110 5.4 The chromaticity of complete multipartite graphs (I) . 113 5.5 The chromaticity of complete multipartite graphs (II) . 119 6 The Chromaticity of Multipartite Graphs 125 6.1 Introduction . 125 6.2 Some basic results . 126 6.3 The chromaticity of tripartite graphs (I) . 128 6.4 The chromaticity of tripartite graphs (II) . 130 6.5 The chromaticity of 4-partite graphs . 137 6.6 The chromaticity of t-partite graphs . 148 Bibliography 155 Notation 163 Index 165 Summary 167 Curriculum Vitae 169 Chapter 1 Introduction For the purpose of tackling the four-color problem, Birkhoff in 1912 [2] in- troduced the chromatic polynomial of a map M, denoted by P (M; ¸), which is the number of proper ¸-colorings of a map M. If one could prove that P (M; 4) > 0 for all maps M, then this would give a positive answer to the four-color problem. Later, Birkhoff and Lewis [3] obtained some results con- cerning the distribution of real roots of chromatic polynomials of planar graphs and conjectured that these polynomials have no real roots greater than or equal to four. The conjecture remains open. In 1932, Whitney [76] generalized the notion of a chromatic polynomial to that of an arbitrary graph and gave many fundamental results for its chromatic polynomial. In 1968, Read [65] asked whether it is possible to find a set of necessary and sufficient algebraic conditions for a polynomial to be the chromatic polynomial of some graph. In particular, Read asked for a necessary and sufficient condition for two graphs to be chromatically equivalent; that is, to have the same chromatic polynomial. In 1978, Chao and Whitehead [8] defined a graph to be chromatically unique if no other graphs share its chromatic polynomial. They gave several families of chromatically unique graphs [9, 10, 11]. Since then many researchers have been studying chromatic uniqueness and chromatic equivalence of graphs. Various families and results on chromatic uniqueness and chromatic equiva- lence of graphs have been obtained successively. The question on chromatic equivalence and uniqueness is called the chromaticity problem of graphs. Very 1 2 Chapter 1 recently, Dong, Koh and Teo finished a monograph on chromatic polynomials and chromaticity of graphs. So, this remains an active area of research. The reader can find more details on the chromaticity of graphs in the survey papers [17, 43, 44, 58, 68] and the monograph [23]. All chromatically unique graphs with 7 vertices and with 8 vertices can be found in [46]. In this thesis, the main aim is to study the algebraic properties of adjoint polynomials and the chromaticity of some classes of graphs. In Section 1.1, we introduce some basic definitions and terminology. In the rest of this introduc- tion we describe our main results, together with some older results. Most of the terminology and notation used in this thesis is standard and can be found in the books [1] by Biggs and [6] by Bondy and Murty. 1.1 Basic definitions and knowledge All graphs considered here are finite and simple. Let V (G), E(G), p(G), q(G) and G denote the vertex set, the edge set, the number of vertices, the number of edges and the complement of a graph G, respectively. For a vertex v of a graph G, we denote by NG(v) the set of vertices of G which are adjacent to v and by d(v) the degree of v. For an edge e = v1v2 of G, set NG(e) = NG(v1) [ NG(v2) ¡ fv1; v2g and d(e) = dG(e) = jNG(e)j.
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