The Fulkerson conjecture states that every bridgeless cubic graph has a collection of six perfect matchings such that each edge belongs to exactly two of them. Mazzuoccolo (2011) showed that this is equivalent to Berge's conjecture that every bridgeless cubic graph should have perfect matching cover index at most 5.
The Fulkerson conjecture is a dual form of the cycle double cover conjecture. Cubic graphs with perfect matching cover index at least 5 are therefore important test objects for this and related conjectures (Máčajová and Škoviera 2021).
The now-refuted Petersen coloring conjecture would have implied the Fulkerson conjecture. Its counterexamples do not refute this weaker conjecture.
