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 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 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
34âm, 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
34âm+2.Comment: 17 pages, 8 figure