Smallest snarks with oddness 4 and cyclic connectivity 4 have order 44

The family of snarks -- connected bridgeless cubic graphs that cannot be 3-edge-coloured -- is well-known as a potential source of counterexamples to several important and long-standing conjectures in graph theory. These include the cycle double cover conjecture, Tutte's 5-flow conjecture, Fulkerson's conjecture, and several others. One way of approaching these conjectures is through the study of structural properties of snarks and construction of small examples with given properties. In this paper we deal with the problem of determining the smallest order of a nontrivial snark (that is, one which is cyclically 4-edge-connected and has girth at least 5) of oddness at least 4. Using a combination of structural analysis with extensive computations we prove that the smallest order of a snark with oddness at least 4 and cyclic connectivity 4 is 44. Formerly it was known that such a snark must have at least 38 vertices [J. Combin. Theory Ser. B 103 (2013), 468--488] and one such snark on 44 vertices was constructed by Lukot'ka et al. [Electron. J. Combin. 22 (2015), #P1.51]. The proof requires determining all cyclically 4-edge-connected snarks on 36 vertices, which extends the previously compiled list of all such snarks up to 34 vertices [J. Combin. Theory Ser. B, loc. cit.]. As a by-product, we use this new list to test the validity of several conjectures where snarks can be smallest counterexamples.Comment: 21 page

Minimal edge colorings of class 2 graphs and double graphs

A proper edge coloring of a class 2 graph G is minimal if it contains a color class of cardinality equal to the resistance r(G) of G, which is the minimum number of edges that have to be removed from G to obtain a graph which is Δ(G)-edge colorable, where Δ(G) is the maximum degree of G. In this paper using some properties of minimal edge colorings of a class 2 graph and the notion of reflective edge colorings of the direct product of two graphs, we are able to prove that the double graph of a class 2 graph is of class 1. This result, recently conjectured, is moreover extended to some generalized double graphs

The Color Number of Cubic Graphs Having a Spanning Tree with a Bounded Number of Leaves

The color number c(G) of a cubic graph G is the minimum cardinality of a color class of a proper 4-edge-coloring of G. It is well-known that every cubic graph G satisfies c(G) = 0 if G has a Hamiltonian cycle, and c(G) ≤ 2 if G has a Hamiltonian path. In this paper, we extend these observations by obtaining a bound for the color number of cubic graphs having a spanning tree with a bounded number of leaves

Generation and Properties of Snarks

For many of the unsolved problems concerning cycles and matchings in graphs it is known that it is sufficient to prove them for \emph{snarks}, the class of nontrivial 3-regular graphs which cannot be 3-edge coloured. In the first part of this paper we present a new algorithm for generating all non-isomorphic snarks of a given order. Our implementation of the new algorithm is 14 times faster than previous programs for generating snarks, and 29 times faster for generating weak snarks. Using this program we have generated all non-isomorphic snarks on $n\leq 36$ vertices. Previously lists up to $n=28$ vertices have been published. In the second part of the paper we analyze the sets of generated snarks with respect to a number of properties and conjectures. We find that some of the strongest versions of the cycle double cover conjecture hold for all snarks of these orders, as does Jaeger's Petersen colouring conjecture, which in turn implies that Fulkerson's conjecture has no small counterexamples. In contrast to these positive results we also find counterexamples to eight previously published conjectures concerning cycle coverings and the general cycle structure of cubic graphs.Comment: Submitted for publication V2: various corrections V3: Figures updated and typos corrected. This version differs from the published one in that the Arxiv-version has data about the automorphisms of snarks; Journal of Combinatorial Theory. Series B. 201

Measurements of edge uncolourability in cubic graphs

Philosophiae Doctor - PhDThe history of the pursuit of uncolourable cubic graphs dates back more than a century. This pursuit has evolved from the slow discovery of individual uncolourable cubic graphs such as the famous Petersen graph and the Blanusa snarks, to discovering in nite classes of uncolourable cubic graphs such as the Louphekine and Goldberg snarks, to investigating parameters which measure the uncolourability of cubic graphs. These parameters include resistance, oddness and weak oddness, ow resistance, among others. In this thesis, we consider current ideas and problems regarding the uncolourability of cubic graphs, centering around these parameters. We introduce new ideas regarding the structural complexity of these graphs in question. In particular, we consider their 3-critical subgraphs, speci cally in relation to resistance. We further introduce new parameters which measure the uncolourability of cubic graphs, speci cally relating to their 3-critical subgraphs and various types of cubic graph reductions. This is also done with a view to identifying further problems of interest. This thesis also presents solutions and partial solutions to long-standing open conjectures relating in particular to oddness, weak oddness and resistance