Generation of cubic graphs and snarks with large girth

TL;DR

This paper introduces two efficient algorithms for generating non-isomorphic cubic graphs and snarks with large girth, significantly improving speed and enabling comprehensive enumeration up to certain sizes.

Contribution

The paper presents novel algorithms for generating cubic graphs and snarks with large girth, achieving substantial speed improvements and enabling enumeration of all such graphs up to specific sizes.

Findings

Algorithms are over 30-40 times faster than previous methods.

Generated all non-isomorphic snarks with girth ≥6 up to 38 vertices.

No snarks with girth ≥7 found up to 42 vertices.

Abstract

We describe two new algorithms for the generation of all non-isomorphic cubic graphs with girth at least $k\ge 5$ which are very efficient for $5\le k \le 7$ and show how these algorithms can be efficiently restricted to generate snarks with girth at least $k$. Our implementation of these algorithms is more than 30, respectively 40 times faster than the previously fastest generator for cubic graphs with girth at least 6 and 7, respectively. Using these generators we have also generated all non-isomorphic snarks with girth at least 6 up to 38 vertices and show that there are no snarks with girth at least 7 up to 42 vertices. We present and analyse the new list of snarks with girth 6.