Generation and Properties of Snarks

TL;DR

This paper introduces a faster algorithm for generating all non-isomorphic snarks up to 36 vertices, analyzes their properties, and provides evidence supporting some conjectures while disproving others.

Contribution

It presents a significantly improved algorithm for snark generation and offers comprehensive analysis of their properties up to 36 vertices.

Findings

Cycle double cover conjecture holds for all generated snarks.

Jaeger's Petersen colouring conjecture is validated for these snarks.

Counterexamples found for eight previous conjectures.

Abstract

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.