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

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 |V(G)| − 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 conjecture 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 oddness 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 LISTOFPAPERS

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, submitted. 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 supervisor 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 being 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 permutation 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({v}) when v ∈ V(G).

NG(v) The set of neighbours to a vertex v ∈ 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) \ 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. [n] The set {1, 2, ... , n}. A k The set of all k-subsets of A.

xv Part I

INTRODUCTIONANDBACKGROUND

1 WHYSNARKS?

”To seek it with thimbles, to seek it with care; To pursue it with forks and hope; To threaten its life with a railway-share; To charm it with smiles and soap! ”For the Snark’s a peculiar creature, that won’t be caught in a commonplace way. Do all that you know, and try all that you don’t: Not a chance must be wasted to-day!

— Lewis Caroll, The Hunting of the Snark

introduction

One of the many great mathematical achievements in the later half of the 20th century was the solution by Appel and Haken [1, 2] to the Four Colour Conjecture. The proof was at the time both revolutionary and controversial among mathematicians since it relied heavily on computers. The question whether four colours always suffice to colour the regions of a map in such a way that neighbouring regions receive different colours may sound like simple problem of interest perhaps mainly to map makers, but it turned out to be a very hard problem of great importance. During the more than one hundred years from when the problem first was mentioned in the mid 19th century until it finally was solved in the late 1970s, it was one of the main driving forces behind the rapid development of graph theory, a field which has since proven its usefulness in a variety of sciences such as computer science, physics, biology, linguistics, sociology and chemistry. One early approach to the four colour theorem (4CT) due to Tait [56] was the study of planar cubic graphs, i.e. 3-regular planar graphs. Tait showed that the 4CT is equivalent to the following.

Theorem 1.0.1 (4CT, edge colouring version). Every 3-connected planar cubic graph is 3-edge colourable.

3 4 why snarks?

However, cubic graphs without separating edges that are not 3-edge colourable are quite elusive structures to study. Until the 1970s only four such graphs were known. The ﬁrst (depicted in Figure 1.1) was discovered by the Danish mathematician Julius Petersen in 1898 [47] (see also [4]). The next two were discovered

Figure 1.1: Some different drawings of the Petersen graph.

by Blanuša [5] in 1946 and they both have 18 vertices. The last one is a snark on 210 vertices which was discovered by Descartes (Tutte) [14] in 1948.

Figure 1.2: One of the two Blanuša snarks. It is easy to see how it is formed from two copies of the Petersen graph. why snarks? 5

In 1976 Martin Gardner wrote about these graphs in his Sci- entific American column [24], and he called them snarks after the elusive creature in Lewis Caroll’s famous poem. A snark, according to Gardner, is a non-trivial cubic graph which cannot be properly edge coloured by three colours. For reasons that will be seen later, we mean by non-trivial that the graph is cyclically 4-edge connected and has girth at least 5. Sometimes the term weak snark is used for a cyclically 4-edge connected, non 3-edge colourable cubic graph with girth 4. During the 1970s more snarks were found and the first infinite family was given by Isaacs [34] (more on this in Chapter 5) in 1975. Still with the advent of computers, it was hard to do an exhaustive search for snarks of a given order. It was not until the 1990s all snarks on 28 vertices and less were found [9, 12]. The problem to determine whether a given cubic graph is 3-edge colourable is not easy in general. It was shown to be NP-complete by Holyer, Leven and Galil [32, 41] and good characterisations of such graphs are therefore unlikely to exist.

Theorem 1.0.2 (Holyer [32], Leven and Galil [41]). It is NP- complete to determine whether χ0(G) = ∆(G) or χ0(G) = ∆(G) + 1 for a cubic graph G.

Not only is it hard to determine whether a given cubic graph is a snark, they are also very rare. It was shown by Robinson and Wormald [49] (see also [7]) that asymptotically almost every cubic graph is hamiltonian, and of course hamiltonian cubic graphs are always 3-edge colourable.

Theorem 1.0.3 (Robinson and Wormald [49]). Let f(n) and g(n) denote the number of hamiltonian cubic graphs respectively the total number of cubic graphs on n vertices. Then limn→ f(n)/g(n) = 1.

Snarks are of interest not only to questions related∞ to the 4CT. There are many other classical conjectures in graph theory that can be reduced1 to this family of graphs. One such question is the cycle double cover conjecture [51, 55] which is one of the main topics of this thesis and will be discussed more in detail in Chapter 2. Tutte’s 5-ﬂow conjecture [57] and Fulkerson’s conjecture [23] (see Chapter 3) are two other such questions. The central themes of this thesis are snarks and conjectures that can be reduced to this family of graphs. The thesis consists of three parts. The ﬁrst is a short introduction to snarks and

1 Reduced here means that if any of these conjectures can be shown to be true for the snarks, then that conjecture must also be true for all graphs. 6 why snarks?

some related problems. The second part is a summary of the appended papers and the third part consists of the four appended papers. Paper I and II present an exhaustive computer search of all snarks on at most 36 vertices and a study of these snarks with respect to various properties (see Chapters 5 and 7). As a result of this computer search counterexamples to a number of different conjectures were found. Also a proof that all cubic graphs with sufficiently long cycles have cycle double covers is given (see Chapter 6). Paper III presents a study of a conjecture called the Petersen colouring conjecture for some well known families of snarks. The last paper, Paper IV, presents a new construction of snarks that are far from being 3-edge colourable. This is covered in chapters 3 and 9. Unless stated otherwise all graphs in this thesis are undirected with multiple edges allowed but not loops. The notation used is hopefully for most part standard in modern graph theory. See e.g. [8, 17, 6, 60] for good comprehensive introductions to the field. A more specific introduction to snarks and cycle covers can be found in [61]. 2 CYCLECOVERS

One main motivation for the study of snarks in this thesis is a classic conjecture in graph theory called the Cycle double cover conjecture. In this chapter we will introduce this conjecture and some related ones.

2.1 some basic definitions

By a cycle1 in a graph G, we mean a connected 2-regular subgraph of G. We often associate a cycle with its edge set to simplify the notation. A graph is said to be cyclically k-edge connected if there exists no edge-cut with k edges such that the deletion of that edge cut creates at least two components of which both contain a cycle. The maximum number k such that a graph G is cyclically k-edge connected is called the cyclic connectivity of G and is denoted by λ(G). An even subgraph H of G is a subgraph where all vertices have even degree. The set of all even subgraphs (seen as edge sets) 2 of a graph G is denoted by CG . As with cycles it is notationally convenient to associate an even subgraph with its edge set. The symmetric difference ∆ of two sets is deﬁned as A∆B := (A \ B) ∪ (B \ A). Given sets A1, A2, ... , Ak, we let ∆{A1, ... , Ak} denote A1∆A2∆ . . . ∆Ak (this is well deﬁned since ∆ is associa- tive). A very useful property of even subgraphs (seen as edge sets) is that the symmetric difference between two even subgraphs is an even subgraph.

Proposition 2.1.1. Given a graph G and A1, A2 ∈ CG. Then A1∆A2 ∈ CG.

1 The deﬁnitions of cycles and even subgraphs as given above are quite standard in graph theory today. Sometimes however, especially in the context of cycle double covers, nowhere-zero ﬂows and cycle decompositions, it is common that even subgraphs are called cycles and connected 2-regular subgraphs are called circuits. 2 CG seen as a vector space over GF(2) with the symmetric difference as vector addition is usually called the cycle space of G.

7 8 cycle covers

The proof can be found in any standard text book in graph theory such as e.g. [8, 17].

2.1.1 Uniform cycle covers

A cycle cover of a graph G is a (multi) set D = {C1, C2, ... , Ck} k of cycles such that E(G) = ∪i=1Ci (i.e. every edge is part of at least one cycle). A uniform cycle cover is a cycle cover where every edge is covered by the same number of cycles, and if this number is k we call this a cycle k-cover. If the cycles in a cycle k-cover D can be coloured with m colours in such a way that cycles which have a common edge receive different colours, we say that it is an m-cycle k-cover. Each of the colour classes here can be seen as even subgraphs, so this is equivalent to a cover by m even subgraphs. It is often convenient, in the context of m-cycle k-covers, to view the set of even subgraphs as the cycle k-cover instead of the set of corresponding cycles. This is something we will use frequently in this thesis.

2.1.2 Cycle decompositions

A cycle 1-cover is also called a cycle decomposition. It is easy to characterise which graphs have cycle decompositions. We say that a graph is even if all vertices have even degree. The following simple proposition is sometimes called Veblen’s theorem [58] (see also [4, 8]). Proposition 2.1.2. A graph has a cycle decomposition if and only if the graph is even. If we prescribe some additional properties that the cycle decomposition should have the situation is not as easy as we will see later in this chapter.

2.2 the cycle double cover conjecture

A cycle double cover (CDC) is a multiset of cycles such that every edge is covered by exactly two cycles. A trivially necessary condition for a graph to have a CDC is that it must be bridgeless. A famous conjecture by Seymour [51] and Szekeres [55] asserts that this condition is also sufﬁcient. Conjecture 2.2.1 (Cycle double cover conjecture (CDCC)). Every bridgeless graph has a cycle double cover. 2.2 the cycle double cover conjecture 9

The following conjecture by Celmins [13] strengthen this further by asserting that every bridgeless graph in fact has a 5-CDC.

Conjecture 2.2.2 (5-CDCC). Every bridgeless cubic graph has a 5- CDC.

It is not too hard to see that a minimal counterexample to both these conjectures must be a cubic graph by using the so called vertex splitting lemma (see e.g. [20]) by Fleischner. For cubic graphs, it is also quite easy to see that those which are 3-edge colourable have CDCs. In fact for those graphs something even stronger holds, namely that every even subgraph is contained in some 4-CDC of the graph.

Proposition 2.2.3. Let G be a 3-edge colourable cubic graph and let D be any 2-regular subgraph of G. Then G has 3-CDC. Furthermore G has a 4-CDC where D is one of the colour classes.

Proof. Let {M1, M2, M3} be the perfect matchings that corresponds to the colour classes in the 3-edge colouring of G. Fur- thermore let Ci = E(G) \ Mi (i.e. the complementary 2-factors to the perfect matchings). Then {C1, C2, C3} is a 3-CDC of G and {D, D∆C1, D∆C2, D∆C3} is a 4-CDC of G. Here the symmetric difference is taken with the cycles seen as sets of edges.

It is also straightforward to see that an edge minimal counterexample to the CDCC cannot have cyclic 2- or 3-edge cuts, nor triangles or 4-cycles (see e.g. [26]). We now have the following well known theorem.

Theorem 2.2.4. A minimal counterexample to the CDCC and 5- CDCC must be a snark.

The converse of Proposition 2.2.3 also holds.

Theorem 2.2.5. Let G be a bridgeless cubic graph with a 2-regular subgraph D. Then the following are equivalent:

(i) G has a 3-CDC.

(ii) G has a 4-CDC which contains D.

(iii) χ0(G) = 3.

Proof. (i) ⇒ (ii) and (iii) ⇒ (i) follows from the proof of Proposition 2.2.3. So it sufﬁces to show (ii) ⇒ (iii). Assume that {C1, C2, C3, C4} are the four even subgraphs in a 4-CDC. 10 cycle covers

Then {C1∆C2, C1∆C3, C1∆C4} are the even subgraphs that corresponds to the colour classes in a 3-CDC. Note that if {D1, D2, D3} are the even subgraphs that corresponds to the colour classes in a 3-CDC, then D1, D2, D3 must be 3|V(G)| 2-factors since |E(G)| = 2 and E(Di) < |V(G)| if Di is not a 2- factor. The 3-edge colouring is now given by the complementary perfect matchings to the 2-factors. From theorems 2.2.5 and 1.0.2 we have the following corollary: Corollary 2.2.6. It is NP-complete to determine if a cubic graph has a k-CDC where k < 5. It is possible to show that snarks which are ”almost” 3-edge colourable (more on this in Chapter 3) also have CDCs. The oddness o(G) of a cubic graph G is the minimal number of odd cycles in any 2-factor3 of G. It is straightforward to see that o(G) = 0 if and only if χ0(G) = 3. Theorem 2.2.7 (Häggkvist and McGuinness [30]). Every bridgeless cubic graph G with o(G) 6 4 has a CDC. One of the main results of Paper I (see Chapter 6) is that bridgeless cubic graphs with sufﬁciently long cycles have CDCs. Theorem 2.2.8 (Hägglund and Markström). Every bridgeless cubic graph which contains a cycle C with |C| > |V(G)| − 9 has a CDC. It is also known that a minimal counterexample must satisfy some stronger conditions such as having large girth. Theorem 2.2.9 (Huck [33]). A minimal counterexample to the CDCC must be a snark with girth at least 12.

2.3 strong cycle double cover conjecture

By Proposition 2.2.3, every 2-regular subgraph of a 3-edge colourable cubic graph G is part of some CDC of G. The ana- loguous statement is however not true in general for bridgeless cubic graphs. For instance, it is straightforward to verify that a 2-factor in the Petersen graph (see Figure 2.1) is not part of any CDC. Unlike the case with even subgraphs, it is still not known if there exists some bridgeless graph with a cycle that cannot be part of a CDC. It was conjectured by Goddyn [28] that no such graph exist:

3 This is not necessary well deﬁned since if the graph has bridges it might not have any 2-factors at all. Therefore it is convenient to extend the deﬁnition by letting o(G) = if G has no 2-factor.

∞ 2.3 strong cycle double cover conjecture 11

Figure 2.1: The bold edges induces a 2-factor in the Petersen graph which cannot be extended to a CDC. Note that all 2-factors in the Petersen graph are isomorphic.

Conjecture 2.3.1 (Strong Cycle Double Cover Conjecture (SCDCC)). Let G be a bridgeless graph. Then for every cycle C in G there is a CDC which contains C.

As with the CDCC it is possible to say something about the structure of a minimal counterexample to the SCDCC.

Deﬁnition 2.3.2. A cycle C in a graph is said to have an extension if there exists another cycle D 6= C such that V(C) ⊆ V(D). If C has no extensions, then C is said to be a stable cycle. A graph which contains stable cycles is said to be a stable graph.

See e.g. [39] for a more detailed proof of the following proposition.

Proposition 2.3.3. Let G be an edge minimal counterexample to SCDCC and C a cycle which cannot be extended to a CDC of H. Then G is a snark and C is a stable cycle.

Proof sketch. First reduce the problem to cubic graphs using the vertex splitting lemma. Then it is straightforward to see that a minimal counterexample cannot have any triangles or cyclic 3-edge cuts. 4-cycles require some case analysis but is also straightforward. Now by Proposition 2.2.3 G cannot be 3-edge colourable and is therefore a snark. Assume that C has an extension D and consider H = G − (E(C) \ E(D)). H is now a 2-connected graph which contains D. This implies that there is a cycle cover of H that covers E(D) 12 cycle covers

once and all other edges twice. By adding the cycles from C∆D we get a cycle cover of G that covers the edges of C once and the rest twice.

For a long time is was not known whether there exists snarks with stable cycles. The ﬁrst construction was given by Kochol [39] and more examples are given in papers I and II (see chapters 6 and 7). In [18] Esteva and Jensen showed that a stronger condition must holds for the cycles of a minimum counterexample to the SCDCC.

Deﬁnition 2.3.4. Let C be a cycle in a graph G. We say that C has a semiextension if there exists another cycle D 6= C in G such that for every path P = xv1v2 . . . vky where x ∈ V(C) \ V(D), y ∈ V(C) ∪ V(D) and v1, ... , vk ∈ V(G) \ (V(C) ∪ V(D)) we have another path P 0 with endpoints x and y where P 0 ⊂ C∆D.A cycle with no semiextension is called a superstable cycle.

Theorem 2.3.5 (Esteva and Jensen [18]). Let (G, C) be a minimal counterexample to SCDCC. Then C must be a superstable cycle.

Esteva and Jensen [18] also posed the following conjecture, which implies the SCDCC.

Conjecture 2.3.6. There are no 2-connected cubic graphs with superstable cycles.

2.3.1 Strong 5-cycle double cover conjecture

An even stronger conjecture than the SCDCC was given by Hoffmann-Ostenhof who conjectured that every cycle is contained in some 5-CDC.

Conjecture 2.3.7 (Hoffmann-Ostenhof [31]). Every cycle C in a bridgeless cubic graph G is contained in some 5-CDC of G.

He also gives a characterisation of this property, which is useful for proving the SCDCC for certain families of snarks: Given a 2-valent vertex v with neighbours u1 and u2 in a graph G, we call that the operation of deleting v from G and adding the edge u1u2 a suppressing of v. Given an edge set A ⊂ E(G), let G ∗ A denote the graph obtained from G − A by suppressing all 2-valent vertices. 2.3 strong cycle double cover conjecture 13

Theorem 2.3.8 (Hoffmann-Ostenhof [31] ). Let G be a cubic graph and D be a 2-regular subgraph of G. Then there is a 5-CDC which contains D if and only if G contains two 2-regular subgraphs D1 and D2 such that D ⊂ D1, M = E(D1) ∩ E(D2) is a matching and G ∗ M is 3-edge colourable.

3 PERFECTMATCHINGCOVERS

Perfect matchings in regular graphs are closely related to edge colourings. For r-regular class 1 graphs1 an r-edge colouring is equivalent to a 1-factorisation of the graph. It is well known that every edge in a bridgeless cubic graph can be part of a perfect matching and therefore every such graph can be covered by perfect matchings (i.e. the union of the perfect matchings equals the edge set of the graph).

Theorem 3.0.9 (Petersen [46]). Let G be a bridgeless cubic graph. Then for every edge e ∈ E(G) there exists a perfect matching M such that e ∈ M.

A consequence of Edmonds matching polytope theorem is that every bridgeless cubic graph also has a uniform covering by perfect matchings (see e.g. [52]). A 3-edge colouring is the same as a 1-covering of the edges by three perfect matchings. But what is the situation for non 3-edge colourable bridgeless cubic graphs?

3.1 fulkerson’s conjecture

Given a graph G and k ∈ N, let kG denote the graph formed by replacing every edge with k multiedges. Fulkerson conjectured that a cubic graph cannot be too far from being 3-edge colourable in the following sense: Given a bridgeless cubic graph G then 2G is always a class 1 graph. Equivalently in terms of coverings by perfect matchings:

Conjecture 3.1.1 (Fulkerson’s conjecture [23, 52]). Every bridgeless cubic graph has six perfect matchings such that every edge is a member of exactly two of them.

Such a cover by six perfect matchings is usually called a Fulkerson cover of G. As with the CDCC it is not so hard to see that a minimal counterexample to this conjecture must be a snark.

1 A graph G is class 1 if χ0(G) = ∆(G).

15 16 perfect matching covers

By considering the complementary 2-factors to the 6 perfect matchings in a Fulkerson cover, it is easy to see that this conjecture has an equivalent formulation in terms of cycle covers. Conjecture 3.1.2 (Fulkerson’s conjecture, complementary version). Every bridgeless cubic graph has a 6-cycle 4-cover. If we allow one more colour class in the cycle 4-cover, such a covering always exist. Theorem 3.1.3 (Bermond et al. [3]). Every bridgeless cubic graph has 7-cycle 4-cover.

3.2 uncolourability measures

There are various ways of measuring how far a snark is from being 3-edge colourable. In this section, we will discuss three such ”measures”.

3.2.1 Perfect matching index

The perfect matching index τ(G) of a cubic graph G is deﬁned as the minimum number of perfect matchings needed to cover E(G). So τ(G) = 3 if and only if χ0(G) = 3 for a cubic graph G. The term perfect matching index was introduced in [22], but other people have studied this concept before (see e.g. [13]). An implication of Fulkerson’s conjecture is that τ(G) 6 5 for every bridgeless cubic graph G. This is known as Berge’s conjecture (see [52]). Conjecture 3.2.1 (Berge). Let G be a bridgeless cubic graph, then τ(G) 6 5. Recently Mazzuoccolo showed that these two conjectures are in fact equivalent [43]. Theorem 3.2.2 (Mazzuoccolo [43]). Fulkerson’s conjecture is equivalent to Berge’s conjecture. Not many partial results on these conjecture are known. For instance, it is not known if there exists a constant k such that τ(G) < k for all snarks G. It is not hard to see that τ(P) = 5 where P is the Petersen graph. A suggested stronger version of Fulkerson’s conjecture is the following: Conjecture 3.2.3 ([44, 22]). Let G be a cyclically 4-edge connected cubic graph. If τ(G) = 5, then G is the Petersen graph. 3.2 uncolourability measures 17

This conjecture is however false and two counterexamples are presented in Paper IV (see also Figure 9.1 and 9.3).

3.2.2 Perfect matchings with empty intersection

Given a cubic graph G, let τ0(G) denote the minimum number k ∈ N such that there is a set {M1, ... , Mk} consisting of Tk k perfect matchings in G with the property that i=1 Mi = ∅. Obviously τ0(G) = 2 if and only if χ0(G) = 3. A weaker assertion than Fulkerson’s conjecture is that every bridgeless cubic graph 0 G has τ (G) 6 3. This is known as the Fan Raspaud Conjecture (FRC).

Conjecture 3.2.4 (Fan and Raspaud [19]). Every bridgeless cubic graph has a set of three perfect matchings, {M1, M2, M3}, such that M1 ∩ M2 ∩ M3 = ∅.

A further weakening of the conjecture was suggested by Berge (see [22]) which is also still open.

Conjecture 3.2.5. There exists k ∈ N such that every bridgeless 0 cubic graph G has τ (G) 6 k. The Fan Raspaud conjecture is equivalent to the statement that every bridgeless cubic graph has a 3-cycle cover where each colour class consists of a 2-factor (consider the complementary 2-factors to the perfect matchings).

Conjecture 3.2.6 (FRC complementary version). Let G be a bridgeless cubic graph. Then G contains three 2-factors F1, F2, F3 such that E(F1) ∪ E(F2) ∪ E(F3) = E(G).

It was pointed out by Goddyn [27] that it is possible to modify the proof of Theorem 3.1.3 to show that it is always possible to ﬁnd a 3-cycle cover where one colour class is a 2-factor. It is also possible to show that cubic graphs which contain two 2-factors such that their union is connected can be covered by two 2-factors and one even subgraph. Given a spanning tree T in a connected graph and an edge e which is not part of T, we let Ce,T denote the edge set of the unique cycle obtained by adding e to T. Furthermore for some subgraph H of a graph G we let H denote the set E(G) \ E(H). The following theorem is well known and the proof can be found in e.g. [8]. 18 perfect matching covers

Theorem 3.2.7. Given a connected graph G with a spanning tree T and a set S ⊂ T we have that C = ∆{Ce,T : e ∈ T} induces an even subgraph of G and C ∩ T = S.

Let G be a cubic graph which contains 2-factors F1 and F2 such that F1 ∪ F2 is connected. Then F1 ∪ F2 contains a spanning T tree and by Theorem 3.2.7 there exists a 2-regular subgraph C such that T ⊂ C. Therefore {F1, F2, C} corresponds to a 3-cycle cover. Furthermore, if T is a Hamiltonian path, then by the following 0 lemma we have τ (G) 6 3. Lemma 3.2.8. Given a cubic graph G with a hamiltonian path P, then C = ∆{Ce,T : e ∈ P} induces a 2-factor. Proof. By Theorem 3.2.7 we know that C induces a 2-regular subgraph of G and P ⊂ C. Since P is a hamiltonian path and every vertex of P is incident with an edge which is not part of P we have that C corresponds to a 2-factor.

A stronger2 result was shown by Máˇcajová and Škoviera [42]. They proved that if a bridgeless cubic graph G has oddness 2, 0 then τ (G) 6 3.

3.2.3 Oddness and resistance

The oddness of a (bridgeless) cubic graph was deﬁned in Chapter 2 as the minimum number of odd cycles in any 2-factor. A related concept introduced by Steffen [54] is the resistance, which is sometimes useful to give bounds for the oddness.

Deﬁnition 3.2.9. The resistance r3(G) of a cubic graph G is de- 0 ﬁned as r3(G) := min{|M| : M ⊂ E(G) and χ (G − M) = 3}, i.e. the minimum number of edges that needs to be removed from G in order to obtain a 3-edge colourable graph.

The relationship between the oddness and resistance is illus- trated in the following proposition:

Proposition 3.2.10. For every cubic graph G it holds that r3(G) 6 o(G).

Proof. If G has no 2-factor then it is trivially true, so assume that G has a 2-factor with k odd cycles. Then by removing one edge from each odd cycle a 3-edge colourable graph is

2 Note that the existence of a hamiltonian path in a cubic graph implies that the oddness of that graph if at most 2. 3.2 uncolourability measures 19

obtained (colour the even cycles alternating between colour 1 and 2, the same with the paths that corresponds to the odd cycles minus one edge, and ﬁnally colour the complementary perfect matching with colour 3).

This is later used in paper IV.

4 PETERSENCOLOURINGSANDFLOWS

In [37] Jaeger studied nowhere-zero flow problems and noted that many of the previously mentioned conjectures in this thesis can be expressed in terms of such flows where the flow values are restricted to certain subsets of some abelian group. Given a set A ⊂ V(G), we let ∂G(A) denote the set of edges with exactly one end in A. For v ∈ V(G) we use ∂G(v) as the short form of ∂G({v}). Let A be an abelian group and B ⊂ A such that 0 6∈ B and B = −B. Then a graph G is said to have a nowhere-zero B-flow if there exists a mapping γ : E(G) → B such that for all S ⊂ V(G) we have that e∈∂S γ(e) = 0. Jaeger observed that if B ⊂ Z5, such that every vector in B has P 2 exactly two 1s, then a graph G has a B-flow if and only if it has 6 a 5-CDC. If on the other hand B ⊂ Z2, such that every vector in B has exactly four 1s, then G has a B-flow if and only if G has a Fulkerson cover. He also noted that both the cycle double cover conjecture and Fulkerson’s conjecture can be unified in a stronger conjecture, which if true would say much of the structure of bridgeless cubic graphs. This unifying conjecture is called the Petersen flow conjecture [37] which also has a useful colouring formulation for cubic graphs.

4.1 petersen colourings

Given graphs G and H, we say that f : E(G) → E(H) is a H- colouring of G if it is a proper edge colouring and for every 0 0 v ∈ V(G) there exists a v ∈ V(H) such that f(∂G(v)) ⊆ ∂H(v ). That is, adjacent edges in G are mapped to adjacent edges in H. Let P denote the Petersen graph. Then the Petersen ﬂow conjecture can be expressed in terms of P-colourings. Conjecture 4.1.1 (The Petersen Colouring Conjecture (PCC) [37]). Every bridgeless cubic graph has a P-colouring. Theorem 4.1.2 (Jaeger [37]). The Petersen colouring conjecture implies the CDCC and Fulkerson’s conjecture. There are a number of other conjectures which are also im- plied by Conjecture 4.1.1. Many of them are conjectures related

21 22 petersen colourings and flows

to graph colourings with points from a Steiner triple system as colours [40]. Another, often more convenient, way of looking at P-colourings, is as what is called normal edge colourings. Let G be a cubic graph and φ : E(G) → {1, 2, 3, 4, 5} be a proper 5-edge coloring. An edge e = xy in G is said to be poor if |φ(∂(x)) ∪ φ(∂(y))| = 3 and it is said to be rich if |φ(∂(x)) ∪ φ(∂(y))| = 5. Let Rφ(G) be the set of rich edges and Pφ(G) be the set of poor edges. Then φ is said to be a normal 5-edge coloring if Rφ(G) ∪ Pφ(G) = E(G) and a strong 5-edge colouring if Rφ(G) = E(G).

Theorem 4.1.3 (Jaeger [36]). A cubic graph has a P-colouring if and only if it has a normal 5-edge colouring.

Proof. Let [n] denote the set {1, 2, 3, ... , n}. It is well known that [5] the P can be deﬁned by V(P) = 2 and E(P) = {vivj : vi ∩ vj = ∅}. For every edge e = xy, let φ(e) := [5] \ (x ∪ y). Then φ is a strong 5-edge-colouring of P (see Figure 4.1). Let G be a bridgeless cubic graph. Assume that G has a P- colouring γ and let η := φ ◦ γ. Consider an edge e = vivj ∈ E(G). Then we have two possibilities: Either γ(∂G(vi)) = γ(∂G(vj)) which would imply that |η(∂G(vi)) ∪ η(∂G(vj))| = 3 and therefore e is a poor edge, or γ(∂G(vi)) ∩ γ(∂G(vj)) = {e} and since every edge in P is rich, we have that |η(∂G(vi)) ∪ η(∂G(vj))| = 5 so e is a rich edge. Hence, η is a normal 5-edge-colouring of G. Conversely, assume that G has a normal 5-edge-colouring η. [5] Note that φ(∂P(vi)) 6= φ(∂P(vj)) if i 6= j and 3 = {φ(∂P(v)) : v ∈ V(P)}. Then it is possible to ﬁnd a mapping f : V(G) → V(P) such that η(∂G(v)) = φ(∂P(f(v)) for v ∈ V(G). For all edges e = xy ∈ E(G) let γ(e) := f(x) ∩ f(y) if e is rich and if e is 0 0 poor let γ(e) := e where e is the edge in ∂P(f(x)) for which φ(e0) = η(e). Then γ is a P-colouring of G.

An equivalent formulation of the Petersen colouring conjecture is therefore:

Conjecture 4.1.4. Every bridgeless cubic graph has a normal 5-edge colouring.

In Paper III it is shown that some families of snarks such as the Flower snarks and the generalised Blanuša snarks satisfy the PCC. 4.1 petersen colourings 23

Figure 4.1: The Petersen graph with a strong 5-edge colouring.

5 FINDINGSNARKS

One of the main obstacles when searching for snarks is that they are very rare among the cubic graphs. Theorem 1.0.3 states that almost all cubic graphs are not just 3-edge colourable but also hamiltonian. Another obstacle is that it is hard to recognise whether a given bridgeless cubic graph is 3-edge colourable or not since the decision problem is NP-complete (see Theorem 1.0.2). In this chapter we will describe some different methods of constructing snarks and also brieﬂy discuss how to do an exhaustive search for snarks of a given order.

5.1 construction of snarks

As mentioned in the introduction only four snarks were known before the 1970s, when Isaacs gave the first constructions of infinite families of snarks. One such family is the Flower snarks discovered by Isaacs [34] in 1974, and according to Watkins [59], also independently by Grinberg in 1972 (but he never published his results). The flower snarks are formed from an odd number of K1,3s connected in a circular manner. The first three members of this family are depicted in Figure 5.1. Another infinite family was discovered by Goldberg [29] in 1981 and is depicted in Figure 5.2.

5.1.1 Isaacs’ dot product

The following lemma (see e.g. [34, 45, 54]) is very useful when studying edge colourability of cubic graphs.

Lemma 5.1.1 (Parity lemma). Let φ : E(G) → {1, 2, 3} be a 3-edge- coloring of a cubic graph G. Then for every edge cut M in G we have that |φ−1(1) ∩ M| ≡ |φ−1(2) ∩ M| ≡ |φ−1(3) ∩ M| ≡ |M|( mod 2)

Given a graph G and v ∈ V(G) we let NG(v) denote the set of vertices in G which share a common edge with v. A consequence of Lemma 5.1.1 is the following theorem which can be used to construct a large family of snarks. The following

25 26 finding snarks

Figure 5.1: The Flower snarks J5, J7 and J9 by Isaacs [34].

construction is usually refereed to as the dot product construction (see Figure 5.3).

Theorem 5.1.2 (Isaacs [34]). Let G1 and G2 be two snarks with edges

e1 = x1x2, e2 = x3x4 ∈ E(G1), f = v1v2 ∈ E(G2) and NG2−f(v1) =

{u1, u2}, NG2−f(v2) = {u3, u4}. Then the cubic graph obtained by deleting e1 and e2 from G1 and the vertices v1 and v2 from G2 and then adding the edges {x1u1, x2u2, x3u3, x4u4} is a snark. A much more general construction for snarks called superposition was discovered by Kochol [38] and can be used to construct snarks with arbitrary high girth. A method somewhat similar to the superposition is described in Paper IV (see also Chapter 9). Other constructions of snarks can be found in e.g. [39, 45, 54, 53, 12].

5.2 generation of snarks

The methods for constructing snarks mentioned in Section 5.1 and also in Paper IV can not be used to construct all snarks of a given order. For that purpose other methods must be used. One such exhaustive search method is described in Paper II (see also Chapter 7). 5.2 generation of snarks 27

Figure 5.2: The Goldberg snarks G5, G7 and G9 [29].

Figure 5.3: An illustration of Isaac’s dot product.

Part II

SUMMARYOFPAPERS

6 PAPERI

Paper I is based on a computer study of the SCDCC for small snarks. At the time public available list of snarks only existed up to 28 vertices. By using the program minibaum [10] by Brinkmann we managed to generate all snarks on 32 vertices and less.

6.1 main results

It is computationally costly to test whether all cycles in a given cubic graph can be part of a CDC. By using the fact that an edge minimal counterexample to the SCDCC must be a snark with a stable cycle (see Proposition 2.3.3 on page 11), the problem can be reduced to only testing the few stable cycles in the snarks that had them. The following observation is made using this reduction: Observation 6.1.1. The SCDCC holds for every snark on 32 vertices and less. Theorem 6.1.2. Every bridgeless cubic graph which contains a cycle C such that |C| > |V(G)| − 9 has a CDC. A simple generalisation of the proof of Theorem 6.1.2 yields the following theorem: Theorem 6.1.3. If k ∈ N and the SCDCC holds for all cubic graphs of order at least 4k then every cubic graph of order n with circumference at least n − k − 1 has a CDC. This is later used in Paper II to give a slightly better bound than that of Theorem 6.1.2. The study of stable cycles among the small snarks also led to two other new results. By starting with a non-dominating stable cycle in a certain snark on 28 and Isaacs’ dot-product (Theorem 5.1.2) we show the following theorem: Theorem 6.1.4. There exists an inﬁnite family of snarks with stable cycles of length 24. In his paper [39] Kochol asked if there exist cyclically 5-edge connected stable snarks. We give a positive answer to that question.

31 32 paper i

Observation 6.1.5. There are four stable cyclically 5-edge connected snarks on 32 vertices and there are no such snarks on fewer vertices.

By using the reduction by Esteva and Jensen (see Theorem 2.3.5 on page 12), Observation 6.1.1 could also be made without explicitly ﬁnding the CDCs for every stable cycle.

Observation 6.1.6. No snark on 32 vertices or less has superstable cycles.

6.2 future work

Theorem 6.1.3 can most likely be improved by using more re- ﬁned methods. However this approach will always give a bound of the type n − k, where n is the number of vertices in the graph and k is some constant, on the length of the cycles needed in order to guarantee a CDC. In Paper IV a construction of an inﬁnite family of snarks where no such constant exists is given. One question that arises naturally from the proof of Theorem 6.1.3 is the following: Problem 1. Are there any snarks which contain induced dominating cycles? The construction used for the proof of Theorem 6.1.4 can easily be used to produce snarks with many disjoint stable cycles. It is not well studied how semiextensions behave in that type of graphs. 7 PAPERII

Paper II is a natural continuation of Paper I. It was apparent that minibaum was insufficient for generating all snarks on any order above 34 vertices (and 34 vertices would most likely require many core decades on a modern cpu). Minibaum is however not a dedicated snark generator. It basically generates all non-isomorphic cubic graph of a given order with some girth and connectivity requirements and then applies an edge colouring filter in the end. This is a very inefficient approach for larger snarks since they are very rare among the cubic graphs. Based on the methods in [11] a program called Snarkhunter1 was developed by Brinkmann and Goedgebeur for generating all non-isomorphic cubic snarks of a given order. It uses a look- ahead see if the graph will be 3-edge colourable by keeping track of the even 2-factors in the graph. This enables pruning of the search tree. Snarkhunter and a parallel computer were used to generate all snarks on 34 and 36 vertices. All weak snarks on 34 vertices and less were also generated.

7.1 main results

7.1.1 Enumeration and properties

The number of snarks compared to the total number of cubic graphs is given in Table 7.1. Enumeration of snarks with respect to certain properties such as girth, cyclic connectivity, hypo- hamiltonicity, automorphism group size and circumference are also given in the paper. The SCDCC was veriﬁed for all snarks on 36 vertices and less and therefore by Theorem 6.1.3, every bridgeless cubic graph with a cycle of length |V(G)| − 10 has a CDC.

1 Snarkhunter is released under GNU General Public License (GPL) and can be downloaded from http://caagt.ugent.be/cubic/.

33 34 paper ii

vertices snarks cubicgraphs ratio

10 1 19 0.0526 12 0 85 0 14 0 509 0 16 0 4060 0 18 2 41 301 4.84 ∗ 10−5 20 6 510 489 1.17 ∗ 10−5 22 20 7 319 447 2.73 ∗ 10−6 24 38 117 940 535 3.22 ∗ 10−7 26 280 2 094 480 864 1.34 ∗ 10−7 28 2900 40 497 138 011 7.16 ∗ 10−8 30 28 399 845 480 228 069 3.36 ∗ 10−8 32 293 059 18 941 522 184 590 1.54 ∗ 10−8 34 3 833 587 453 090 162 062 723 8.46 ∗ 10−9 36 60 167 732 11 523 392 072 541 432 5.22 ∗ 10−9

Table 7.1: The number of snarks compared to the total number of cubic graphs. The number of cubic graphs is obtain from Gordon Royle’s homepage [50] (see also Robinson and Wormald [48]).

A number of conjectures, such as the SCDCC, orientable 5- CDCC and the Petersen colouring2 conjecture were veriﬁed to hold for the snarks on 36 vertices and less. The strong 5-CDCC was also tested for all snarks on 34 vertices and less, and for the cyclically 5-edge connected snarks on 36 vertices.

7.1.2 Counterexamples

One approach to proving the CDCC proposed by Fleischner et al. [21] was to consider compatible cycle decompositions (see Paper II for deﬁnitions) of well connected 4-regular graphs. This is equivalent to a problem concerning a special kind of cubic graphs called permutation graphs. A permutation graph is a cubic graph which contains a 2-factor consisting of exactly two cycles, and each of these cycles is an induced subgraph (i.e. they have

2 The PCC was also veriﬁed for the weak snarks on 34 vertices and less since we were unable to show that a minimal counterexample cannot have any 4-cycles. 7.1 main results 35 no chords). A snark which is also a permutation graph is called a permutation snark. The approach is basically based on the hope that the Petersen graph would be the only cyclically 5-edge connected permutation snark with a 2-factor that cannot be part of a CDC. This is however not true. The snarks depicted in Figure 7.1 and 7.2 provides direct counterexamples to the following four conjectures (see Paper II for more deﬁnitions and how the conjectures are related to each other).

Conjecture 7.1.1 (Zhang [61]). Let G be a cubic cyclically 5-edge- connected permutation graph. If G is a snark, then G must be the Petersen graph.

Conjecture 7.1.2 (Jackson [35], Zhang [61]). Let G be a cyclically 5-edge-connected cubic graph and D be a set of pairwise disjoint cycles of G. Then D is a subset of a CDC, unless G is the Petersen graph.

Conjecture 7.1.3 (Zhang [61]). Let G be a permutation graph with the chordless cycles C1 and C2 where C1 ∪ C2 is a 2-factor. If G is cyclically 5-edge-connected and there is no CDC which includes both C1 and C2, then G must be the Petersen graph.

Conjecture 7.1.4 (Zhang [62]). Let G be a bridgeless cubic graph and w be an eulerian (1, 2)-weight of G where e∈T w(e) > 4 for every cyclic edge-cut T. Then if (G, w) has no compatible cycle cover, P then G must be the Petersen graph.

The graphs in in Figure 7.1 and 7.2 also provide indirect counterexamples to the following three conjectures.

Conjecture 7.1.5 (Fleischner et al. [21]). Let G be a connected 4-regular graph and T be a transition system. Then (G, T) has no compatible cycle decomposition if and only if (G, T) ∈ R.

Conjecture 7.1.6 (Fleischner et al. [21]). If G is an essentially 6-edge-connected 4-regular graph with a transition system T, then (G, T) has no compatible cycle decomposition if and only if (G, T) is the bad loop or the bad K5.

Conjecture 7.1.7 (Genest [25]). The only Sabidussi orbits that are pure, prime and circle orbits are the orbit of the white isolated vertex and the orbit of the white pentagon. 36 paper ii

Figure 7.1: Six of the cyclically 5-edge connected permutation snarks on 34 vertices.

7.2 future work

It is probably computationally feasible to generate all snarks on 38 vertices as well, but to do an exhaustive search for all snarks on 40 vertices will most likely not be possible within the next few years. It would be of interest to construct more specialised snark generators for e.g. snarks of high cyclic connectivity and permutation snarks. In [16] Jaeger and Swart conjectured that no snark has cyclic edge-connectivity more than 6. The smallest snark of girth 7 is still not known, but considering that snarks of smallest order with girth 5 and 6 were cyclically 5- respectively 6-edge connected, it is not a too wild guess that the smallest snark with girth 7 is actually cyclically 7-edge connected. Work is currently being done to characterise the structure of the counterexamples depicted in Figure 7.1 and 7.2. A result of 7.2 future work 37

Figure 7.2: The other six cyclically 5-edge connected permutation snarks on 34 vertices. this is also a construction of an inﬁnite family of cyclically 5- edge connected permutation snarks where the deﬁning 2-factor is not part of a CDC.

8 PAPERIII

The seed for this paper came from a short breakfast discussion in the beautiful slovenian town Bled during the 7th Slovenian International Conference on Graph Theory 2011. Surprisingly little work has been done on the Petersen colouring conjecture, with the notable exception of the work by Jaeger [37, 36] and DeVos et al. [15].

8.1 main results

The first two results in the paper concerns a property called strong Petersen colourings. Intuitively this can be seen as a Pe- tersen colouring which also preserves the orientation on the cycles (see Paper IV for the formal definition). A cubic graph which is strongly Petersen colourable also has an orientable 5-CDC and a nowhere zero 5-flow. Theorem 8.1.1. Every generalised Blanuša snark of type 1 with an odd number of A-blocks is strongly Petersen colourable.

Theorem 8.1.2. For every k > 1, the ﬂower snark J2k+1 is not strongly Petersen colourable. A theorem on the structure of a minimal counterexample to the PCC is also given. Theorem 8.1.3. If G is a minimum counterexample to the Petersen colouring conjecture (or to the Fulkerson conjecture), then it does not ∗ contain K3,3 as a subgraph (see Figure 8.1). Theorem 8.1.3 can also be used to prove that every ﬂower snark has a Petersen colouring.

Theorem 8.1.4. For all k > 1, the ﬂower snark J2k+1 has a Petersen colouring. The last two results in the paper are that two structured families of snarks have Petersen colourings.

Theorem 8.1.5. Every Goldberg snark Gk, where k > 5 is odd, has a Petersen colouring. Theorem 8.1.6. All generalised Blanuša snarks of type 1 and 2 have Petersen colourings.

39 40 paper iii

∗ Figure 8.1: The graph K3,3. A minimal counterexample to the PCC cannot have this as a subgraph.

8.2 future work

It was shown by Jaeger [36] that a minimal counterexample to the PCC must be a cyclically 4-edge connected cubic graph girth at least 4 (i.e. a weak snark). It would be of great interest to investigate if it is possible to show that a minimal counterexample cannot have any cyclic 4-edge cuts (and therefore also no 4-cycles). However this is most likely very hard and a more reasonable aim might be to at least show that it cannot have any 4-cycles. 9 PAPERIV

A snark G is said to be a strong snark if for every v ∈ V(G) it holds that χ0(G ∗ {v}) = 4. It was observed in Paper II that the graph depicted in Figure 9.1 is the most symmetric of the strong snarks with a minimum number of vertices. It soon became clear that the graph has more interesting properties and Paper IV is an attempt to characterise some of the structure in that graph and generalise it.

Figure 9.1: A strong snark on 34 vertices with some interesting properties.

9.1 main results

Two inﬁnite families of snarks are constructed in Paper IV. The constructions are based on the fact that any cubic graph which contains either the graph H1 or H2 (see Figure 9.2) as subgraphs cannot be 3-edge colourable. The snarks in this family have arbitrary high oddness and resistance. Both these invariants grow linearly in the number vertices. The circumference graph G is the length of its longest cycle and we denote this by circ(G). The following simple lemma

41 42 paper iv

(a) The graph H1

(b) The graph H2

Figure 9.2: Two non 3-edge colourable graphs with ﬁve vertices of degree 2 and all other with degree 3.

states that the graphs constructed in Paper IV also have short circumference.

Lemma 9.1.1. Let G be a cubic graph. Then circ(G) 6 |V(G)| − r3(G) + 1. At least two of the constructed snarks also turned out to have some other interesting properties. The following conjecture has been suggested as a possible strengthening of Fulkerson’s conjecture. Conjecture 9.1.2 ([44, 22]). Let G be a cyclically 4-edge connected cubic graph. If τ(G) = 5, then G is the Petersen graph. However the following observation shows that this conjecture is false. Observation 9.1.3. There exists a snark G distinct from the Petersen graph with τ(G) = 5 and where no 2-factor can be part of a CDC. The counterexample is the graph in Figure 9.1. Another counterexample is given by the graph in Figure 9.3 which is also obtained from the construction in the paper. It is however not veriﬁed that no 2-factor in this graph can be part of a CDC (but it is true that no 2-factor can be part of 5-CDC). 9.2 future work 43

Figure 9.3: A strong snark on 46 vertices constructed from the prism and a 4-cycle. This graph requires at least 5 perfect matching for a covering of the edges.

9.2 future work

It is very unsatisfying only having a computer veriﬁcation of the fact that the snarks depicted in Figure 9.1 and 9.3 have perfect matching index 5. A non-computer proof might lead to better insight regarding that property and perhaps also a construction of an inﬁnite family of such snarks.

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Part III

PAPERSI-IV