On Sylvester Colorings of Cubic Graphs

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 states that for any bridgeless cubic graph G, one has: P ≺ G. The second author has recently introduced the Sylvester coloring conjecture, which states that for any cubic graph G one has: S ≺ G. Here S is the Sylvester graph on ten vertices. In this paper, we prove the analogue of Sylvester coloring conjecture for cubic pseudo-graphs. Moreover, we show that if G is any connected simple cubic graph G with G ≺ P , then G = P . This implies that the Petersen graph does not admit an S16-coloring, where S16 is the Sylvester graph on 16 vertices. Finally, we show that any cubic graph G has a coloring with edges of Sylvester graph S such that at least 4 5 of vertices of G meet the conditions of Sylvester coloring conjecture.