H-colorings for 4-regular graphs

For a vertex a in H, we denote by partial differential 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 ( partial differential G (u)) = partial differential 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. (c) 2023 Elsevier B.V. All rights reserved.