On Cubic Bridgeless Graphs Whose Edge-Set Cannot be Covered by Four Perfect Matchings
Louis Esperet, Giuseppe Mazzuoccolo · Journal of Graph Theory · 2013
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 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 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 . We also construct the first known snark with no cycle cover of length less than .