A tris of perfect matchings in bridgeless claw-free cubic graphs

Abstract

A proof of the cycle double cover conjecture was recently announced, yielding an $8$-cycle double cover for every bridgeless graph. The stronger $5$-cycle double cover conjecture, which is still open, is equivalent to the statement that the edge set of every bridgeless claw-free cubic graph can be covered by at most four perfect matchings. Perfect matchings in bridgeless cubic graphs have been studied extensively, with two of the main conjectures in this area being the Berge--Fulkerson and the Fan--Raspaud conjectures. The latter, a consequence of the former, states that every bridgeless cubic graph admits three perfect matchings $M_1, M_2, M_3$ such that $M_1\cap M_2\cap M_3=\emptyset$. Here we show that the Fan--Raspaud conjecture is true for bridgeless claw-free cubic graphs. This also gives further information on the interaction of perfect matchings in a class where the $5$-cycle double cover conjecture requires control of four of them.