Generation of Cycle Permutation Graphs and Permutation Snarks

TL;DR

This paper introduces an efficient algorithm to generate all non-isomorphic cycle permutation graphs and permutation snarks up to certain sizes, providing new bounds and insights into their properties.

Contribution

It presents a novel algorithm for generating cycle permutation graphs and permutation snarks, extending previous computational results and addressing open questions in graph theory.

Findings

Generated all cycle permutation graphs up to order 34.

Generated all permutation snarks up to order 46.

Provided new lower bounds for specific permutation snarks.

Abstract

We present an algorithm for the efficient generation of all pairwise non-isomorphic cycle permutation graphs, i.e. cubic graphs with a $2$-factor consisting of two chordless cycles, non-hamiltonian cycle permutation graphs and permutation snarks, i.e. cycle permutation graphs that do not admit a $3$-edge-colouring. This allows us to generate all cycle permutation graphs up to order $34$ and all permutation snarks up to order $46$, improving upon previous computational results by Brinkmann et al. Moreover, we give several improved lower bounds for interesting permutation snarks, such as for a smallest permutation snark of order $6 \bmod 8$ or a smallest permutation snark of girth at least $6$ and give more evidence in support of a conjecture of Goddyn. These computational results also allow us to complete a characterisation of the orders for which non-hamiltonian cycle permutation graphs exist, answering an open question by Klee from 1972, and yield many more counterexamples to conjectures by Jackson and Zhang.