Berge-Fulkerson coloring for infinite families of snarks

TL;DR

This paper proves that all graphs in a certain infinite family of snarks, constructed from cyclically 4-edge-connected graphs, have a Fulkerson-cover, confirming a long-standing open problem.

Contribution

It establishes that every graph in the specified infinite family possesses a Fulkerson-cover, solving an open problem in the study of snarks and perfect matchings.

Findings

All graphs in the family have a Fulkerson-cover.

The result applies to an infinite class of snarks.

It confirms the conjecture for this family of graphs.

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.