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Self-Complementary Graphs and Generalisations: a Comprehensive Reference Manual

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Self-Complementary Graphs and Generalisations: a Comprehensive Reference Manual Self-complementary graphs and generalisations: a comprehensive reference manual Alastair Farrugia University of Malta August 1999 ii This work was undertaken in part ful¯lment of the Masters of Science degree at the University of Malta, under the supervision of Prof. Stanley Fiorini. Abstract A graph which is isomorphic to its complement is said to be a self-comple- mentary graph, or sc-graph for short. These graphs have a high degree of structure, and yet they are far from trivial. Su±ce to say that the problem of recognising self-complementary graphs, and the problem of checking two sc-graphs for isomorphism, are both equivalent to the graph isomorphism problem. We take a look at this and several other results discovered by the hun- dreds of mathematicians who studied self-complementary graphs in the four decades since the seminal papers of Sachs (UbÄ er selbstkomplementÄare graphen, Publ. Math. Drecen 9 (1962) 270{288. MR 27:1934), Ringel (Selbstkomple- mentÄare Graphen, Arch. Math. 14 (1963) 354{358. MR 25:22) and Read (On the number of self-complementary graphs and digraphs, J. Lond. Math. Soc. 38 (1963) 99{104. MR 26:4339). The areas covered include distance, connectivity, eigenvalues and colour- ing problems in Chapter 1; circuits (especially triangles and Hamiltonicity) and Ramsey numbers in Chapter 2; regular self-complementary graphs and Kotzig's conjectures in Chapter 3; the isomorphism problem, the reconstruc- tion conjecture and self-complement indexes in Chapter 4; self-complement- ary and self-converse digraphs, multi-partite sc-graphs and almost sc-graphs in Chapter 5; degree sequences in Chapter 6, and enumeration in Chapter 7. This is a manual more than a survey, as it contains all the results I could ¯nd, and quite a few proofs. There are also a few original results. For any queries, comments or suggestions, please contact me at afarrugia at alumni dot uwaterloo dot ca. iii iv Contents Sec. Pg. Acknowledgements 0.1 ix Updates and Correspondence 0.2 x Notation and references 0.3 x Layout of the thesis 0.5 xii What was left out, and what wasn't 0.7 xiv A word of warning 0.13 xvi 1 Introduction and Fundamental Properties 1 Distance and connectivity 1.4 3 Degree sequences and antimorphisms 1.21 11 New graphs from old 1.26 13 Auto- and anti-morphisms 1.29 16 The structure of sc-graphs 1.36 21 Planarity and thickness 1.46 27 Eigenvalues 1.48 28 Chromaticity 1.57 32 The Hadwiger and Strong Perfect Graph Conjectures 1.64 36 The chromatic and total chromatic index 1.67 38 More Nordhaus-Gaddum results 1.74 41 2 Circuits, Paths and Cliques 45 Triangle numbers and paths 2.3 46 Circuits and Hamiltonicity 2.11 50 Circuits and Hamiltonicity in sc-tournaments 2.28 58 Ramsey Numbers 2.31 59 3 Regular Self-Complementary Graphs 63 Two interlocking hierarchies 3.3 64 Paley digraphs and generalisations 3.18 76 Characterisation of some vertex-transitive sc-digraphs 3.29 81 Strongly regular sc-graphs 3.32 82 Kotzig's problems 3.37 85 Antimorphisms with unequal cycle lengths 3.43 91 Open Problems 3.46 95 v 4 Self-Complementarity 97 The isomorphism problem 4.2 97 Generating self-complementary graphs 4.11 104 Self-complement indexes 4.19 111 Logical connections 4.32 118 The reconstruction conjectures 4.34 120 Codes and information 4.43 127 5 Generalisations of Self-Complementary Graphs 131 Self-complementary and self-converse digraphs 5.2 131 Multipartite self-complementary graphs 5.13 141 r-partite antimorphisms 5.16 142 Bipartite self-complementary graphs 5.21 145 Paths in r-partite sc-graphs 5.25 148 Diameters in r-psc-graphs 5.28 149 Almost self-complementary graphs 5.35 152 Isomorphic decompostitions 5.45 159 Cyclically t-complementary graphs 5.51 163 \Super-symmetrical" graphs 5.60 167 6 Degree Sequences 169 Self-complementary graphs and digraphs 6.2 170 Edge degree sequences 6.9 173 Bipartite sc-graphs and sc-tournaments 6.18 177 Open problems 6.24 181 7 Enumeration 183 De¯nitions 7.2 184 Self-complementary graphs and digraphs 7.7 186 Prime order vertex-transitive digraphs 7.18 192 Relations and self-converse digraphs 7.26 198 Special types of graphs and digraphs 7.32 201 Colour cyclic factorisations 7.42 207 Self-complementary k-plexes 7.55 211 Self-complementary orbits and functions 7.58 214 Open problems 7.64 217 Bibliography 219 vi List of Figures 1.1 The small sc-graphs . 2 1.2 H + K K and H + K P . 6 2 ± 1 2 [ 4 1.3 Sc-graphs of diameter 3 without end-vertices . 8 1.4 A sc-graph of order 4k + 4 and diameter 3 without end-vertices 15 1.5 A sc-graph of order 4k + 5 and diameter 3 without end-vertices 15 1.6 The orbits of a sc-graph; the bold lines denote the edges of N(G). 20 1.7 The structure of sc-graphs . 26 1.8 The largest self-complementary line graph . 31 1.9 The graphs Un, n = 4; 5; 8; 9 . 34 2.1 Sc-graphs with no 2-factor . 56 Regular self-complementary graphs | two interlocking hierarchies . 65 3.2 Cay(Z ; 1; 3; 6; 7; 10; 12 ) . 88 13 f g 3.3 G9 . 89 3.4 Harts¯eld's graph . 92 3.5 H(k) . 94 4.1 SD1;D2 and SD1;D2 . 100 4.2 SD1 and SD2 . 101 4.3 (G; H) and (G; H) . 102 P P 4.4 (G; G) and (H; H) . 102 P P 4.5 A graph with s(G) = k, k 2 or 3 (mod 4) . 113 ´ 5.1 The self-converse digraphs of order 3 . 132 5.2 The self-complementary digraphs of order 3 . 133 5.3 A self-converse digraph with no involutory antimorphism . 135 5.4 A sc-tournament with trivial automorphism group . 136 vii 5.5 A bipsc graph with only mixed antimorphisms . 146 5.6 J1 and J2 . 146 5.7 Bipsc-graphs with isolated vertices . 147 5.8 A path of length 6 and its bipartite complement . 149 5.9 The almost self-complementary graphs on 6 vertices; the spe- cial vertices are drawn in black. 153 5.10 Two vertices at distance 4 in G~. 156 viii Acknowledgements 0.1. Writing this thesis has been like working on a four-hundred piece jigsaw puzzle, except that instead of houses and trees, the pieces contained theorems and equations, sometimes in di®erent languages and very di®erent notation, not to mention that they were (literally) scattered around the globe. For helping me to gather some of the pieces, I must thank all those who generously sent reprints of their papers (and sometimes even other people's papers):- Michael Albertson, Giorgio Balconi, Lowell Beineke, Paul Camion, ² Gary Chartrand, Charles Colbourn, Tamonas Gangopadhyay, Fr¶ed¶eric Havet, A.J.W. Hilton, Bill Kocay, Jan Kratochv¶³l, Flavia Mamanna, Terry McKee, Ronald Read, Chris Rodger, Alexander Rosa, Ivo Rosen- berg, Norma Salvi-Zagaglia, Allen Schwenk, A. Vijayakumar, Earl Glen Whitehead, Jr. and A. Pawel Wojda; Stanley Fiorini who obtained other papers, and lent me the invaluable ² four-volume Graph Theory Reviews and other books; Irene Sciriha, who brought to my attention Stockmeyer's papers on the reconstruction conjecture and lent me various books; and Josef Lauri who gave me photocopies of other papers. Gino Favero who sent me photocopies of over 50 papers which were not ² available in Malta; and Susan Raimondo and Daniele Varacca, who sent other photocopies; the journals section of the University of Malta Library, whose value is ² sometimes questioned by people who have never done any research. I would also like to thank the following people for their help: my employers, Magus Int.l, and especially Jackie Chetcuti, Mario Borg ² and Arnold Stellini, for granting and then extending more than a week's leave at point blank notice, so that I could ¯nish my thesis Peter Wiosna, who translated parts of H. Sachs' \UbÄ er selbstkomple- ² mentÄare graphen", the ¯rst paper on self-complementary graphs; those who provided information: Edward Bender, Luigi Borzacchini, ² Gary Chartrand, Christopher Clapham, Gouli Ding, Luis Goddyn, Bill ix Kocay, Brendan McKay, Rhys Price Jones, Erich Prisner, Mukesh Pun- hani, Igor Rivin, Ivo Rosenberg, Allen Schwenk, Jozef Sir¶· an· and Detlef Wille; those who helped me get in touch with other people: Anne Bres (sec- ² retary at the Institut National de Recherche en Informatique et en Au- tomatique), Dimitri Chernyak, Vladimir Chernyak, Mike J. Grannell, Terry Griggs, Fausta Guzzoni (secretary at the University of Parma), Jo Ann McFarland (secretary at the University of California), Steve Quinn, Catherine P. Vistro-Yu, C.R. Rao, John Sheehan and Robin J. Wilson; and sta® at the Department of Maths for helping in various ways | ² James L. Borg, who provided lots of technical assistance; Joseph Mus- cat and the secretary, Annabel Attard. I am very grateful to Brian Alspach for suggesting Juraj Bos¶ak's \De- compositions of graphs" when I was still starting on my research. It is an excellent survey book which I have often referred to both for content and style; and from which I ¯rst learnt that there is more to self-complementary graphs than the \n 0 or 1 (mod 4)" lemma. I thank Prof. Fiorini´ for his patience and help; and my parents for their support. Last, but de¯nitely not least, I would like to thank my sister Nadine who drew and re-drew (and re-drew1) the diagrams.
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