Berge–Fulkerson coloring for C(8)‐linked graphs
Rong‐Xia Hao, Cun‐Quan Zhang, Ting Zheng · Journal of Graph Theory · 2017
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. Let G be a cubic graph and be a 2‐factor of G such that is odd if and only if for some integer k. The 2‐factor F is C(8)‐linked if, for every , there is a circuit of length 8 with edge sequence where and . And the cubic graph G is C(8)‐linked if it contains a C(8)‐linked 2‐factor. It is proved in this article that every C(8)‐linked cubic graph is Berge–Fulkerson colorable. It is also noticed that many classical families of snarks (including some high oddness snarks) are C(8)‐linked. Consequently, the Berge–Fulkerson conjecture is verified for these infinite families of snarks.