On two consequences of Berge–Fulkerson conjecture

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. The first of them states that for any bridgeless cubic graph G, edge e and i with 0≤i≤2, there are three perfect matchings F1,F2,F3 of G such that F1∩F2∩F3=0̸ and e belongs to exactly i of these perfect matchings. The second one states that for any bridgeless cubic graph G and its vertex v, there are three perfect matchings F1,F2,F3 of G such that F1∩F2∩F3=0̸ and the edges incident to v belong to k1, k2 and k3 of these perfect matchings, where the numbers k1, k2 and k3 satisfy the obvious necessary conditions. In the paper, 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.