Berge-Fulkerson coloring for infinite families of snarks

It is conjectured by Berge and Fulkerson that every bridgeless cubic graph has six perfect matchings such that each edge is contained in exactly two of them. Hägglund constructed two graphs Blowup(K4, C) and Blowup(Prism,C4). Based on these two graphs, Chen constructed infinite families of bridgeless cubic graphs M0,1,2,...,k−2,k−1 which is obtained from cyclically 4-edge-connected and having a Fulkerson-cover cubic graphs G0, G1, . . . , Gk−1 by recursive process. If each Gi for 1 ≤ i ≤ k − 1 is a cyclically 4-edge-connected snarks with excessive index at least 5, Chen proved that these infinite families are snarks. He obtained that each graph in M0,1,2,3 has a Fulkersoncover and gave the open problem that whether every graph in M0,1,2,...,k−2,k−1 has a Fulkerson-cover. In this paper, we solve this problem and prove that every graph in M0,1,2,...,k−2,k−1 has a Fulkerson-cover. Keyword: Berge-Fulkerson conjecture, Fulkerson-cover, perfect matching. MSC(2010): Primary: 05C70; Secondary: 05C75, 05C40, 05C15.