Perfect Matching Covers of Cubic Graphs of Oddness 2

A perfect matching cover of a graph G is a set of perfect matchings of G such that each edge of G is contained in at least one member of it. Berge conjectured that every bridgeless cubic graph has a perfect matching cover of order at most 5. The Berge Conjecture is largely open and it is even unknown whether a constant integer c does exist such that every bridgeless cubic graph has a perfect matching cover of order at most e. In this paper, we show that a bridgeless cubic graph G has a perfect matching cover of order at most 11 if G has a 2-factor in which the number of odd circuits is 2.