Let G be a bridgeless cubic graph. The Berge-Fulkerson Conjecture (1970s)
states that G admits a list of six perfect matchings such that each edge of
G belongs to exactly two of these perfect matchings. If answered in the
affirmative, two other recent conjectures would also be true: the Fan-Raspaud
Conjecture (1994), which states that G admits three perfect matchings such
that every edge of G belongs to at most two of them; and a conjecture by
Mazzuoccolo (2013), which states that G admits two perfect matchings whose
deletion yields a bipartite subgraph of G. It can be shown that given an
arbitrary perfect matching of G, it is not always possible to extend it to a
list of three or six perfect matchings satisfying the statements of the
Fan-Raspaud and the Berge-Fulkerson conjectures, respectively. In this paper,
we show that given any 1+-factor F (a spanning subgraph of G such that
its vertices have degree at least 1) and an arbitrary edge e of G, there
always exists a perfect matching M of G containing e such that
G∖(F∪M) is bipartite. Our result implies Mazzuoccolo's
conjecture, but not only. It also implies that given any collection of disjoint
odd circuits in G, there exists a perfect matching of G containing at least
one edge of each circuit in this collection.Comment: 13 pages, 8 figure