H-colorings for 4-regular graphs

Abstract

For a vertex a in H, we denote by ∂H(a) the set of edges incident with a. An H-coloring of a graph G is a proper edge-coloring f:E(G)→E(H) such that for each vertex u in G, there exists a vertex a in H such that f(∂G(u))=∂H(a). In this paper, we introduce some graphs H such that H-colorability of 4-regular graphs is meaningful and related to some of their properties, such as an even cycle decomposition of size 3 and an even 2-factor. In addition, we formulate a conjecture on H-coloring of the line graph of some cubic graphs. Partial solutions to the conjecture are also presented.