On two consequences of Berge–Fulkerson conjecture

The classical Berge–Fulkerson conjecture states that any bridgeless cubic graph Gadmits a list of six perfect matchings such that each edge of Gbelongs 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 eand iwith 0≤i≤2, there are three perfect matchings F1,F2,F3of Gsuch that F1∩F2∩F3=0̸and ebelongs to exactly iof these perfect matchings. The second one states that for any bridgeless cubic graph Gand its vertex v, there are three perfect matchings F1,F2,F3of Gsuch that F1∩F2∩F3=0̸and the edges incident to vbelong to k1, k2and k3of these perfect matchings, where the numbers k1, k2and k3satisfy 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.