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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 , 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 .

On Cubic Bridgeless Graphs Whose Edge‐Set Cannot be Covered by Four Perfect Matchings