A note on [formula omitted]-coloring and [formula omitted]-coloring 4-regular graphs

Abstract

Let ∂H(u) be the set of edges incident with a vertex u in the graph H. We say that a graph G is H-colorable if there exist total functions f:E(G)→E(H) and g:V(G)→V(H) such that f is a proper edge-coloring of G and for each vertex u∈V(G) we have f(∂G(u))=∂H(g(u)). Let X¯ be the graph obtained by adding three parallel edges between two degree one vertices of the graph K1,4. Let Aˆ be the graph obtained by adding two pendant edges to two different vertices of a triangle and then adding two edges between the degree two vertex and the two adjacent degree three vertices. Malnegro and Ozeki (2024) [7] asked whether every 4-regular graph with an even number of vertices and an even cycle decomposition admits an X¯-coloring or an Aˆ-coloring and whether every 2-connected planar 4-regular graph with an even number of vertices admits such a coloring. Additionally, they conjectured that for every 2-edge-connected simple cubic graph G with an even number of edges, the line graph L(G) is X¯-colorable. In this short note, we discuss two algorithms for deciding whether a graph G is H-colorable. We give negative answers to the two questions and disprove the conjecture by finding suitable graphs, as verified by two independent algorithms.