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Graph Partitions and the Bichromatic Number Dennis D.A. Epple Doctor of Philosophy

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Graph Partitions and the Bichromatic Number Dennis D.A. Epple Doctor of Philosophy Graph Partitions and the Bichromatic Number by Dennis D.A. Epple Diplom, Freie Universit¨atBerlin, 2005 A Dissertation Submitted in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy in the Department of Mathematics and Statistics c Dennis D.A. Epple, 2011 University of Victoria All rights reserved. This dissertation may not be reproduced in whole or in part by photocopy or other means, without the permission of the author. Graph Partitions and the Bichromatic Number by Dennis D.A. Epple Diplom, Freie Universit¨atBerlin, 2005 Supervisory Committee Dr. Jing Huang, Supervisor (Department of Mathematics and Statistics) Dr. Peter Dukes, Member (Department of Mathematics and Statistics) Dr. Gary MacGillivray, Member (Department of Mathematics and Statistics) Dr. Frank Ruskey, Outside Member (Department of Computer Science) ii Supervisory Committee Dr. Jing Huang, Supervisor (Department of Mathematics and Statistics) Dr. Peter Dukes, Member (Department of Mathematics and Statistics) Dr. Gary MacGillivray, Member (Department of Mathematics and Statistics) Dr. Frank Ruskey, Outside Member (Department of Computer Science) Abstract A(k; l)-colouring of a graph is a partition of its vertex set into k independent sets and l cliques. The bichromatic number χb of a graph is the minimum r such that the graph is (k; l)-colourable for all k + l = r. The bichromatic number is related to the cochromatic number, which can also be defined in terms of (k; l)-colourings. The bichromatic number is a fairly recent graph parameter that arises in the study of extremal graphs related to a classical result of Erd}os,Stone and Simonovits, and in the study of the edit distance of graphs from hereditary graph classes. While the cochromatic number has been well studied in the literature, there are only few known structural results for the bichromatic number. A main focus of this thesis is to establish a foundation of knowledge about the bichromatic number. The secondary focus is on (k; l)-colourings of certain interesting graph classes. Two known bounds for the bichromatic number are χb ≤ χ + θ − 1, where χ is p the chromatic number and θ the clique covering number of the graph, and χb ≥ n, where n the number of vertices of the graph. We give a complete characterization of all graphs for which equality holds in the first bound, and show that the second bound is best possible by constructing graphs for square numbers n such that equality holds in the bound. We investigate graphs for which the bichromatic number equals the cochromatic number and prove a Brooks-type theorem for the bichromatic number. iii Regarding (k; l)-colourings, we find a new algorithm for calculating the (k; l)- colourability of cographs and show that cographs have a particularly nice represen- tation with regard to (k; l)-colourings. For proper circular arc graphs, we provide a method for (k; l)-colouring if l ≥ 1, and establish an algebraic characterization for all maximally (k; 0)-colourable proper circular arc graphs. Finally, we investigate the bichromatic number and cochromatic with respect to lexicographic products and show several nice bounds. iv Contents Supervisory Committee ii Abstract iii Table of Contents v List of Tables viii List of Figures ix Acknowledgments xii 1 Introduction 1 1.1 Colouring variations . 1 1.2 Special graph classes . 3 1.3 Other topics . 6 1.4 Terminology . 7 1.5 Glossary of notation . 9 2 Covering graphs with independent sets and cliques 12 2.1 (k; l)-colouring . 14 2.1.1 The colouring sequences κ and λ . 16 2.1.2 Small graphs that do not have a (k; l)-colouring . 21 2.1.3 Complexity . 25 2.2 The cochromatic number . 26 2.2.1 Basic properties . 27 v 2.2.2 Complexity . 28 2.3 The bichromatic number . 29 2.3.1 Basic properties . 29 2.3.2 Complexity . 34 3 The bichromatic number for general graphs 36 3.1 χb in terms of χ and θ ......................... 38 3.1.1 Cographs . 38 3.1.2 Box cographs . 39 3.1.3 Characterization of graphs with χb = χ + θ − 1 . 43 3.2 Square graphs . 50 3.3 Graphs with χb = χc .......................... 57 3.4 A Brooks-type theorem for χb ..................... 63 4 Special graph classes 76 4.1 Cographs . 78 4.1.1 Cotrees . 84 4.1.2 Basic algorithms on cotrees . 88 4.1.3 Cotrees and (k; l)-colouring . 94 4.2 Chordal graphs . 100 4.2.1 Perfect elimination ordering and (k; l)-colourings . 100 4.2.2 Bichromatic number of k-trees . 102 4.3 Round digraphs (proper circular arc graphs) . 107 4.3.1 (k; l)-colourings with l ≥ 1 . 110 4.3.2 (k; l)-colourings with l = 0 . 116 4.3.3 Maximally k-colourable round digraphs . 125 4.3.4 Forbidden subgraphs . 137 5 Fractional versions and the lexicographic product 140 5.1 Fractional versions . 142 vi 5.1.1 Fractional (k; l)-colouring . 143 5.1.2 Fractional cochromatic number . 143 5.1.3 Fractional bichromatic number . 146 5.2 Lexicographic product . 151 5.2.1 Classes closed under the lexicographic product . 159 5.2.2 Lexicographic product of particular graphs . 161 Bibliography 165 vii List of Tables Table 2.1 Values satisfying the inequalities in Proposition 2.1.19. 25 Table 5.1 Examples for comparabilities among the colouring parameters. 150 viii List of Figures Figure 1.1 P4 and C5............................. 6 Figure 2.1 A (3; 0)-colouring, a (1; 2)-colouring and a (0; 3)-colouring of a graph. 15 Figure 2.2 The pairs (k; l) for which the graph is not (k; l)-colourable. 16 Figure 2.3 Young diagram of κ(G), where G is the graph from Figure 2.1. 18 Figure 2.4 Young diagrams of κ(G) = (3; 3; 1), κ(H) = (3; 2; 1; 1) and κ(G _ H) = (6; 5; 2; 1). 21 Figure 2.5 The bichromatic and cochromatic numbers of the graph from Figure 2.1. 30 Figure 2.6 The Gr¨otzsch graph. 32 Figure 3.1 A box cograph of dimension 3 times 4. 39 Figure 3.2 (r − 1; s − 1)-colouring G by parts. 44 Figure 3.3 (k; l)-colourings with k + l = 4. 51 Figure 3.4 An affine plane of order 3 with its parallel classes. 52 Figure 3.5 An affine plane graph with the various (k; 3 − k)-colourings indicated. 53 Figure 3.6 The graph P4........................... 53 Figure 3.7 Slope graph of order 9. 54 Figure 3.8 Various (k; 4 − k)-colourings of a slope graph of order 16. 55 Figure 3.9 The graph Q. .......................... 65 ix Figure 4.1 A (3; 2)-colouring of a cograph given Young diagram repre- sentation. 81 Figure 4.2 Young diagram representation of G1, G2 and G1 + G2. 82 Figure 4.3 A cotree. 85 Figure 4.4 The cograph corresponding to the cotree from Figure 4.3. 86 Figure 4.5 A pseudocotree of the graph from Figure 4.4. 86 Figure 4.6 CHROMATIC NUMBER on the cotree from Figure 4.3. 89 Figure 4.7 CHROMATIC NUMBER on an induced subgraph contain- ing the vertices indicated by the arrows. 90 Figure 4.8 MAXIMUM CLIQUE on the cotree from Figure 4.3. 93 Figure 4.9 KAPPA for the cotree from Figure 4.3. 95 Figure 4.10 BOX COGRAPH with k = 4 and l = 2 for the cotree from Figure 4.3. 98 Figure 4.11 YOUNG DIAGRAM for the complement of the cotree from Figure 4.3. 99 Figure 4.12 A balanced 2-path. 104 Figure 4.13 A round digraph. 108 Figure 4.14 The essential arcs of the round digraph from Figure 4.13. 109 Figure 4.15 The clique consisting of v0; v3; v6 is maximal but not transitive. 111 Figure 4.16 A 5-colouring of a round digraph given by its essential arcs. 118 Figure 4.17 A consecutive 5-colouring (inside) and a 5-permutation la- belling (outside) of a round digraph given by its essential arcs. 121 Figure 4.18 A path diagram. 127 Figure 4.19 A reduced path diagram. 130 2 Figure 5.1 A fractional (1; 3 )-colouring of K3 + 2K1. 144 Figure 5.2 Relationships among the colouring parameters. 148 x Figure 5.3 Relationships among the colouring parameters for vertex- transitive and perfect graphs. 149 Figure 5.4 The graph P4[P3]. 151 Figure 5.5 A (3; 2)-colouring of P4[P3] and the projection of its sets onto P4. ................................ 154 Figure 5.6 Maximal independent sets and cliques in P4. 162 Figure 5.7 Covering C5 twice with k independent sets and l cliques with k + l = 5. 163 xi Acknowledgements I want to thank my supervisor, Jing Huang. I am grateful for his support and guidance and for challenging me to improve every step of the way. Thanks to all department members at the University of Victoria, who have made the years of my Ph.D. an enjoyable experience. In particular, I want to thank Peter Dukes and Gary MacGillivray for expanding my horizon in Discrete Mathematics, Kieka Mynhardt for always being there, and Pauline van den Driessche, without whom I might not have come to Victoria. Thanks to Stephen Benecke and Kseniya Garaschuk for being fantastic officemates and great friends. Also thanks to Michelle Edwards and all members of Hector's Friends for their friendship. Thanks to Magdalena Georgescu for all that and so much more. My thanks go to Gernot Stroth and Martin Aigner for introducing me to the beauty of Discrete Mathematics, and to Martin Kutz for fuelling my passion for research and for long talks about mathematics.
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