S_12 and P_12-colorings of cubic graphs

Abstract

If G and H are two cubic graphs, then an H -coloring of G is a proper edge-coloring f with the edges of H , such that for each vertex x of G , there is a vertex y of H with f ( ∂ G ( x )) = ∂ H ( y ). If G admits an H -coloring, then we will write H ≺ G . The Petersen coloring conjecture of Jaeger ( P 10 -conjecture) states that for any bridgeless cubic graph G , one has: P 10 ≺ G . The S 10 -conjecture states that for any cubic graph G , S 10 ≺ G . In this paper, we introduce two new conjectures that are related to these conjectures. The first of them states that any cubic graph with a perfect matching admits an S 12 -coloring. The second one states that any cubic graph G whose edge-set can be covered with four perfect matchings, admits a P 12 -coloring. We call these new conjectures S 12 -conjecture and P 12 -conjecture, respectively. Our first results justify the choice of graphs in S 12 -conjecture and P 12 -conjecture. Next, we characterize the edges of P 12 that may be fictive in a P 12 -coloring of a cubic graph G . Finally, we relate the new conjectures to the already known conjectures by proving that S 12 -conjecture implies S 10 -conjecture, and P 12 -conjecture and (5, 2)-Cycle cover conjecture together imply P 10 -conjecture. Our main tool for proving the latter statement is a new reformulation of (5, 2)-Cycle cover conjecture, which states that the edge-set of any claw-free bridgeless cubic graph can be covered with four perfect matchings.