Construction of Permutation Snarks

TL;DR

This paper constructs an infinite family of cyclically 5-edge connected permutation snarks with a specific 2-factor property, and proves that this 2-factor is not contained in any circuit double cover, advancing understanding of snark structures.

Contribution

It introduces a new infinite family of permutation snarks with unique 2-factor properties and demonstrates their circuit double cover limitations.

Findings

Constructed an infinite family of cyclically 5-edge connected permutation snarks.

Proved the permutation 2-factor is not in any circuit double cover for these snarks.

Enhanced understanding of the structural properties of permutation snarks.

Abstract

A permutation snark is a snark which has a 2-factor $F_2$ consisting of two chordless circuits; $F_2$ is called the permutation 2-factor of $G$. We construct an infinite family $\mathcal H$ of cyclically 5-edge connected permutation snarks. Moreover, we prove for every member $G \in \mathcal H$ that the permutation 2-factor given by the construction of $G$ is not contained in any circuit double cover of $G$.