Algorithms for the Generation of Snarks

TL;DR

This paper introduces two new algorithms for generating snarks, a special class of cubic graphs with specific edge-colouring properties, and provides comprehensive lists of these graphs up to certain sizes.

Contribution

The paper presents novel algorithms for generating snarks with specific girth constraints and supplies complete lists of such graphs up to 40 vertices.

Findings

Generated all girth 4 snarks up to 40 vertices

Generated all girth ≥5 snarks up to 38 vertices

Compiled lists of strong snarks up to 40 vertices

Abstract

The essential requirement for a cubic graph to be called a snark is that it can not be edge-coloured with three colours. To avoid trivial cases, varying restrictions on the connectivity are imposed. Snarks are not only interesting in themselves, but also a valuable test field for conjectures about graphs that are not snarks and sometimes not even cubic. For many important open problems in graph theory it is proven that minimal counterexamples would be snarks. We give two new algorithms for the generation of snarks and results of computer programs implementing these algorithms. One algorithm is for snarks with girth exactly 4 and is used for generating complete lists of girth 4 snarks on up to 40 vertices. The second algorithm lists snarks with girth at least 5 and is used for generating complete lists of such snarks on up to 38 vertices. We also give complete lists of strong snarks (in the terminology of Jaeger) on up to 40 vertices.