  In 1994, it was conjectured by Fan and Raspaud that every simple bridgeless
cubic graph has three perfect matchings whose intersection is empty. In this
paper we answer a question recently proposed by Mkrtchyan and Vardanyan, by
giving an equivalent formulation of the Fan-Raspaud Conjecture. We also study a
possibly weaker conjecture originally proposed by the first author, which
states that in every simple bridgeless cubic graph there exist two perfect
matchings such that the complement of their union is a bipartite graph. Here,
we show that this conjecture can be equivalently stated using a variant of
Petersen-colourings, we prove it for graphs having oddness at most four and we
give a natural extension to bridgeless cubic multigraphs and to certain cubic
graphs having bridges.
