On Fulkerson conjecture

TL;DR

This paper explores the Fulkerson conjecture in bridgeless cubic graphs, demonstrating how to derive a Fulkerson covering from two FR-triples and proving the conjecture for certain snark classes.

Contribution

It introduces a method to obtain Fulkerson coverings from FR-triples and proves the conjecture for specific well-known snarks.

Findings

Fulkerson covering can be derived from two FR-triples.

The Fulkerson conjecture holds for some classes of snarks.

Provides a simple proof for certain cases.

Abstract

If $G$ is a bridgeless cubic graph, Fulkerson conjectured that we can find 6 perfect matchings (a{\em Fulkerson covering}) with the property that every edge of $G$ is contained in exactly two of them. A consequence of the Fulkerson conjecture would be that every bridgeless cubic graph has 3 perfect matchings with empty intersection (this problem is known as the Fan Raspaud Conjecture). A {\em FR-triple} is a set of 3 such perfect matchings. We show here how to derive a Fulkerson covering from two FR-triples. Moreover, we give a simple proof that the Fulkerson conjecture holds true for some classes of well known snarks.