  The classical Berge-Fulkerson conjecture states that any bridgeless cubic
graph $G$ admits a list of six perfect matchings such that each edge of $G$
belongs to two of the perfect matchings from the list. In this short note, we
discuss two statements that are consequences of this conjecture. We show that
the first statement is equivalent to Fan-Raspaud conjecture. We also show that
the smallest counter-example to the second one is a cyclically
$4$-edge-connected cubic graph.
