On snarks that are far from being 3-edge colorable

TL;DR

This paper constructs infinite families of snarks with high oddness and low circumference, providing counterexamples to a Fulkerson conjecture strengthening and revealing new properties about cycle double covers.

Contribution

It introduces new infinite snark families with specific properties and offers a counterexample to a proposed Fulkerson conjecture strengthening.

Findings

Constructed two infinite snark families with high oddness and low circumference.

Provided a counterexample to a Fulkerson conjecture strengthening.

Showed the counterexample has no 2-factor in a cycle double cover.

Abstract

In this note we construct two infinite snark families which have high oddness and low circumference compared to the number of vertices. Using this construction, we also give a counterexample to a suggested strengthening of Fulkerson's conjecture by showing that the Petersen graph is not the only cyclically 4-edge connected cubic graph which require at least five perfect matchings to cover its edges. Furthermore the counterexample presented has the interesting property that no 2-factor can be part of a cycle double cover.