A remark on the Petersen coloring conjecture of Jaeger

If G and H are two cubic graphs, then we write H ≺ G if G admits 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). Let P and S be the Petersen graph and the Sylvester graph, respectively. In this paper, we introduce the Sylvester coloring conjecture. Moreover, we show that if G is a connected bridgeless cubic graph with G ≺ P , then G = P . Finally, if G is a connected cubic graph with G ≺ S, then G = S.