3 research outputs found
REDUCTION OF THE BERGE-FULKERSON CONJECTURE TO CYCLICALLY 5-EDGE-CONNECTED SNARKS
The Berge-Fulkerson conjecture, originally formulated in the language of mathematical programming, asserts that the edges of every bridgeless cubic (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-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-edge-connected
An equivalent formulation of the Fan-Raspaud Conjecture and related problems
In 1994, it was conjectured by Fan and Raspaud that every simple bridgeless cubic
graph has three perfect matchings whose intersection is empty. In this paper we answer
a question recently proposed by Mkrtchyan and Vardanyan, by giving an equivalent formulation of the Fan-Raspaud Conjecture. We also study a possibly weaker conjecture
originally proposed by the first author, which states that in every simple bridgeless cubic
graph there exist two perfect matchings such that the complement of their union is a bipartite graph. Here, we show that this conjecture can be equivalently stated using a variant of
Petersen-colourings, we prove it for graphs having oddness at most four and we give a natural extension to bridgeless cubic multigraphs and to certain cubic graphs having bridges.peer-reviewe
Projective, affine and abelian colorings of cubic graphs, submitted
We develop an idea of a local 3-edge-coloring of a cubic graph, a generalization of the usual 3-edge-coloring. We allow for an unlimited number of colors but require that the colors of two edges meeting at a vertex always determine the same third color. Local 3-edge-coloring