Reduction of the Berge-Fulkerson conjecture to cyclically 5-edge-connected snarks

Abstract

The Berge-Fulkerson conjecture, originally formulated in the language of mathematical programming, asserts that the edges of every bridgeless cubic ( 3 3 -valent) graph can be covered with six perfect matchings in such a way that every edge is in exactly two of them. As with several other classical conjectures in graph theory, every counterexample to the Berge-Fulkerson conjecture must be a non- 3 3 -edge-colorable cubic graph. In contrast to Tutte’s 5-flow conjecture and the cycle double conjecture, no nontrivial reduction is known for the Berge-Fulkerson conjecture. In the present paper, we prove that a possible minimum counterexample to the conjecture must be cyclically 5 5 -edge-connected.