Perfect matchings with restricted intersection in cubic graphs

A conjecture of G. Fan and A. Raspaud asserts that every bridgeless cubic graph contains three perfect matchings with empty intersection. We propose a possible approach to this and similar problems, based on the concept of a balanced join in an embedded graph. We use this method to prove that bridgeless cubic graphs of oddness two have Fano colorings using only five lines of the Fano plane. This is a special case of a conjecture by E. Máčajová and M. Škoviera.