Berge–Fulkerson coloring for some families of superposition snarks. (August 2021) - Liu, Siyan; Hao, Rong-Xia; Zhang, Cun-Quan
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. This conjecture has been verified for many families of snarks with small ( ≤ 5 ) cyclic edge-connectivity. An infinite family, denoted by S K, of cyclically 6-edge-connected superposition snarks was constructed in [European J. Combin. 2002] by Kochol. In this paper, the Berge–Fulkerson conjecture is verified for the family S K, and, furthermore, some larger families containing S K . This is the first paper about the Berge–Fulkerson conjecture for superposition snarks and cyclically 6-edge-connected snarks. Tutte's integer flow and Catlin's contractible configuration are applied here as the key methods.