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 12conjecture, 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 12coloring 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.

On the existence of graphs which can colour every regular graph

“…S 4 , and since Hf has a unique vertex of degree 1, it cannot be obtained by splitting unused vertices of some other graph, and so, Hf = H f , as required. To conclude this section we provide a generalisation of the following two theorems, proved in [13] and [5]. Theorem 3.5 (Mkrtchyan, 2013 [13]).…”

“…Then H S 10 . Theorem 3.6 (Hakobyan & Mkrtchyan, 2019 [5]). Let H be a connected cubic graph with H ≺ S 12 .…”

“…The multigraph S 10 is also referred to as the Sylvester graph and is depicted together with the multigraph S 12 in Figure 1 (see also [4]).…”

“…In what follows we make use of some results contained in Lemma 2.2 in [5]. We reproduce only the part of the lemma that we shall need in the sequel, even if in a slightly more general form.…”

An equivalent formulation of the Fan-Raspaud Conjecture and related problems

“…The importance of H-colourings is mainly due to Jaeger's Conjecture [14] which states that each bridgeless cubic graph G admits a P -colouring (where P is again the Petersen graph). For recent results on P -colourings, known as Petersen-colourings, see for instance [12,13,26,29]. The following proposition shows why we choose to refer to a pair of perfect matchings whose deletion leaves a bipartite subgraph as an S 4 -pair.…”