2013/01/31 by Louis Esperet, Giuseppe Mazzuoccolo · 1 citation
Mathematics · #math.CO #msc:05C15
paper · pdf · doi:10.1002/jgt.21778
published as J. Graph Theory 77(2) (2014), 144-157 · 17 pages, 8 figures
arxiv created 2013/11/15 · arxiv updated 2014/09/17
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 \cal 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 \cal 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 \tfrac43m, and we show that this inequality is strict for graphs of \cal F. We also construct the first known snark with no cycle cover of length less than \tfrac43m+2.