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 article, we prove that deciding whether this number is at most four for a given cubic bridgeless graph is NP-complete. We also construct an infinite family F of snarks (cyclically 4-edge-connected cubic graphs of girth at least 5 and chromatic index 4) whose edge-set cannot be covered by four perfect matchings. Only two such graphs were known. It turns out that the family 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 4m/3, and we show that this inequality is strict for graphs of F. We also construct the first known snark with no cycle cover of length less than 4m/3 + 2.