S_12 and P_12-colorings of cubic graphs
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If G and H are two cubic graphs, then an H-coloring of G is a proper edge-coloring f with 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 Sylvester coloring conjecture (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.
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