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

Even cycles with prescribed chords in planar cubic graphs

AbstractThe following result is being proved. Theorem: Let e be an arbitrary line of the 2-connected, 3-regular, planar graph G such that e does not belong to any cut set of size 2. The G contains an even cycle for which e is a chord

A note about the dominating circuit conjecture

AbstractThe dominating circuit conjecture states that every cyclically 4-edge-connected cubic graph has a dominating circuit. We show that this is equivalent to the statement that any two edges of such a cyclically 4-edge-connected graph are contained in a dominating circuit

On circuit decomposition of planar Eulerian graphs

AbstractWe give a common generalization of P. Seymour's “Integer sum of circuits” theorem and the first author's theorem on decomposition of planar Eulerian graphs into circuits without forbidden transitions