Berge-Fulkerson coloring for infinite families of snarks

Abstract: 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. H$\ddot{a}$gglund constructed two graphs Blowup$(K_4, C)$ and Blowup$(Prism, C_4)$. Based on these two graphs, Chen constructed infinite families of bridgeless cubic graphs $M_{0,1,2, \ldots,k-2, k-1}$ which is obtained from cyclically 4-edge-connected and having a Fulkerson-cover cubic graphs $G_0,G_1,\ldots, G_{k-1}$ by recursive process. If each $G_i$ for $1\leq i\leq k-1$ 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 $M_{0,1,2,3}$ has a Fulkerson-cover and gave the open problem that whether every graph in $M_{0,1,2, \ldots,k-2, k-1}$ has a Fulkerson-cover. In this paper, we solve this problem and prove that every graph in $M_{0,1,2, \ldots,k-2, k-1}$ has a Fulkerson-cover.