On cubic bridgeless graphs whose edge-set cannot be covered by four perfect matchings

The problem of establishing the number of perfect matchings necessary to cover the edge-set of a cubic bridgeless graph is strictly related to a famous conjecture of Berge and Fulkerson. In this paper we prove that deciding whether this number is at most 4 for a given cubic bridgeless graph is NP-complete. We also construct an infinite family $\mathcal{F}$ of snarks (cyclically 4-edge-connected cubic graphs of girth at least five and chromatic index four) whose edge-set cannot be covered by 4 perfect matchings. Only two such graphs were known. It turns out that the family $\mathcal{F}$ also has interesting properties with respect to the shortest cycle cover problem. The shortest cycle cover of any cubic bridgeless graph with $m$ edges has length at least $\frac{4}{3}m$, and we show that this inequality is strict for graphs of $\mathcal{F}$. We also construct the first known snark with no cycle cover of length less than $\frac{4}{3}m+2$.