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SNARKS Generation, Coverings and Colourings SNARKS Generation, coverings and colourings jonas hägglund Doctoral thesis Department of mathematics and mathematical statistics Umeå University April 2012 Doctoral dissertation Department of mathematics and mathematical statistics Umeå University SE-901 87 Umeå Sweden Copyright © 2012 Jonas Hägglund Doctoral thesis No. 53 ISBN: 978-91-7459-399-0 ISSN: 1102-8300 Printed by Print & Media Umeå 2012 To my family. ABSTRACT For a number of unsolved problems in graph theory such as the cycle double cover conjecture, Fulkerson’s conjecture and Tutte’s 5-flow conjecture it is sufficient to prove them for a family of graphs called snarks. Named after the mysterious creature in Lewis Carroll’s poem, a snark is a cyclically 4-edge connected 3-regular graph of girth at least 5 which cannot be properly edge coloured using three colours. Snarks and problems for which an edge minimal counterexample must be a snark are the central topics of this thesis. The first part of this thesis is intended as a short introduction to the area. The second part is an introduction to the appended papers and the third part consists of the four papers presented in a chronological order. In Paper I we study the strong cycle double cover conjecture and stable cycles for small snarks. We prove that if a bridgeless cubic graph G has a cycle of length at least jV(G)j - 9 then it also has a cycle double cover. Furthermore we show that there exist cyclically 5-edge connected snarks with stable cycles and that there exists an infinite family of snarks with stable cycles of length 24. In Paper II we present a new algorithm for generating all non- isomorphic snarks with a given number of vertices. We generate all snarks on 36 vertices and less and study these with respect to various properties. We find that a number of conjectures on cycle covers and colourings holds for all graphs of these orders. Furthermore we present counterexamples to no less than eight published conjectures on cycle coverings, cycle decompositions and the general structure of regular graphs. In Paper III we show that Jaeger’s Petersen colouring con- jecture holds for three infinite families of snarks and that a minimum counterexample to this conjecture cannot contain a certain subdivision of K3,3 as a subgraph. Furthermore, it is shown that one infinite family of snarks have strong Petersen colourings while another does not have any such colourings. Two simple constructions for snarks with arbitrary high odd- ness and resistance is given in Paper IV. It is observed that some snarks obtained from this construction have the property that they require at least five perfect matchings to cover the v edges. This disproves a suggested strengthening of Fulkerson’s conjecture. vi LIST OF PAPERS Paper I J. Hägglund and K. Markström. On stable cycles and cycle double covers of graphs with large circumference. Discrete Mathematics (in press). Paper II G. Brinkmann, J. Goedgebeu, J. Hägglund and K. Markström. Generation and properties of snarks, sub- mitted. Paper III J. Hägglund and E. Steffen. Petersen colorings and some families of snarks, submitted. Paper IV J. Hägglund. On snarks that are far from being 3-edge colorable, manuscript. vii By this art you may contemplate the variations of the 23 letters. Anatomy of Melancholy, Pt. 2, Sec. II, Mem. IV — Jorge Luis Borges, The Library of Babel PREFACE When I began my undergraduate studies at Umeå university in 2001 my main field of study was not mathematics but computer science. I do not think I really made any conscious choice to change the direction, but when the time came to choose a topic for my master thesis, I had taken almost equally many courses in both fields. The reason for my shift towards mathematics was certainly not because I found it easier (quite the contrary), but perhaps because mathematics has a beauty and an elegance that is hard to describe. During my time as a PhD student I have had the pleasure of exploring the world of mathematics, not only in an abstract sense, but also physically. I was able to visit and attend courses at universities all around Sweden thanks to the National Gradu- ate School for Scientific Computing (NGSSC) and I had a fan- tastic summer in Berlin thanks to the Research Training Group Methods for Discrete Structures (MDS) which organised a very nice course with the title Random and Quasirandom Graphs at Humboldt-Universität in 2008. I was also able to attend great conferences and workshops in Poland, Germany, Hungary, Den- mark, France, England (twice), Scotland and Slovenia. There are many people to whom I am very grateful for their support during my time as a PhD student. First of all, I would like express my gratitude to my supervi- sor Professor Roland Häggkvist. Thank you for giving me the opportunity to work with you and for generously sharing your extensive knowledge in graph theory. I would also like to thank my assistant supervisor and co-author Klas Markström for our good cooperation and for providing me with many ideas for research. I am grateful to the staff of the department of mathematics and mathematical statistics for being such nice and helpful col- leagues. In particular I would like to thank Margareta Brinkstam, Berith Melander and Ingrid Westerberg-Eriksson for always be- ing cheerful, helping and for making the department a happier place. ix I wish to specially thank Carl Johan Casselgren, Konrad Abramowicz, David Källberg and Magnus Lindh for always being great friends and for all the good times we had together. Thank you also CJ for reading the early versions of this thesis, as well as many of my papers. To all my friends from Byviken and neighbouring villages, thank you for all the fun and crazy things we have done over the years. I wish to thank my family to whom I dedicate this thesis. I thank my parents Erik and Carina for always supporting me and believing in me. I am also grateful to my brother Per and my sisters Emma and Sara. Finally, thank you Yu for all your love, encouragement and support. Thank you all! Jonas Hägglund March 2012 x CONTENTS i introduction and background1 1 why snarks? 3 2 cycle covers7 2.1 Some basic definitions . 7 2.1.1 Uniform cycle covers . 8 2.1.2 Cycle decompositions . 8 2.2 The cycle double cover conjecture . 8 2.3 Strong cycle double cover conjecture . 10 2.3.1 Strong 5-cycle double cover conjecture . 12 3 perfect matching covers 15 3.1 Fulkerson’s conjecture . 15 3.2 Uncolourability measures . 16 3.2.1 Perfect matching index . 16 3.2.2 Perfect matchings with empty intersection 17 3.2.3 Oddness and resistance . 18 4 petersen colourings and flows 21 4.1 Petersen colourings . 21 5 finding snarks 25 5.1 Construction of snarks . 25 5.1.1 Isaacs’ dot product . 25 5.2 Generation of snarks . 26 ii summary of papers 29 6 paper i 31 6.1 Main results . 31 6.2 Future work . 32 7 paper ii 33 7.1 Main results . 33 7.1.1 Enumeration and properties . 33 7.1.2 Counterexamples . 34 7.2 Future work . 36 8 paper iii 39 8.1 Main results . 39 8.2 Future work . 40 9 paper iv 41 9.1 Main results . 41 9.2 Future work . 43 xi xii contents bibliography 45 iii papers i-iv 53 LISTOFFIGURES Figure 1.1 Drawings of the Petersen graph . 4 Figure 1.2 A Blanuša snark . 4 Figure 2.1 A 2-factor in the Petersen graph . 11 Figure 4.1 The Petersen graph with a strong 5-edge colouring. 23 Figure 5.1 The Flower snarks J5, J7 and J9 ....... 26 Figure 5.2 The Goldberg snarks G5, G7 and G9 .... 27 Figure 5.3 Isaac’s dot product . 27 Figure 7.1 The first six cyclically 5-edge connected permutation snarks . 36 Figure 7.2 The other six 5-edge connected permuta- tion snarks . 37 ∗ Figure 8.1 K3,3 ....................... 40 Figure 9.1 A strong snark on 34 vertices . 41 Figure 9.2 Two non 3-edge colourable 5-poles . 42 Figure 9.3 A strong snark on 46 vertices . 43 LISTOFTABLES Table 7.1 Number of snarks . 34 xiii NOTATION E(G) The edge set of a graph G. V(G) The vertex set of a graph G. girth(G) The length of a shortest cycle in G. δ(G) The minimum degree of any vertex in G. ∆(G) The maximum degree of any vertex in G. @G(X) The set of edges with exactly one end in X ⊂ V(G). @G(v) Short form of @G(fvg) when v 2 V(G). NG(v) The set of neighbours to a vertex v 2 V(G) CG The set of all even subgraphs (seen as edge sets) in G. χ0(G) The edge chromatic number of G. λ(G) The cyclic edge-connectivity of G. o(G) The oddness of G. r3(G) The resistance of G. g(G) The length of the shortest cycle in G. circ(G) The length of the longest cycle in G. H The set E(G) n E(H) for a subgraph H ⊂ G. τ(G) The perfect matching index of G. τ0(G) The cardinality of a minimal set of perfect matchings with empty intersection. G ∗ A The graph obtained from G - A by suppressing all 2-valent vertices. kG The graph obtained from G by replacing every edge with k multiedges.
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