A note on $\bar{X}$-coloring and $\hat{A}$-coloring 4-regular graphs
Jorik Jooken
TL;DR
The paper investigates $H$-colorability for $H\in\{\bar{X},\hat{A}\}$ on $4$-regular graphs and related line-graph questions. It introduces a straightforward recursive algorithm to decide $H$-colorability and uses it to derive negative answers to MO24's questions and to refute a conjecture by exhibiting counterexamples, including a planar $4$-regular graph on $18$ vertices and the Tietze graph. Independent verification with a second algorithm confirms the results and provides publicly available code. The findings constrain the landscape of $H$-colorability for $4$-regular graphs and motivate further study of $\hat{A}$-colorability in line graphs, as well as exploration of larger counterexamples.
Abstract
Let $\partial_H(u)$ 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 $f : E(G) \rightarrow E(H)$ and $g : V(G) \rightarrow V(H)$ such that $f$ is a proper edge-coloring of $G$ and for each vertex $u \in V(G)$ we have $f(\partial_G(u))=\partial_H(g(u))$. Let $\bar{X}$ be the graph obtained by adding three parallel edges between two degree one vertices of the graph $K_{1,4}$. Let $\hat{A}$ 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 [Discrete Math. 347(3):113844 (2024)] asked whether every 4-regular graph with an even number of vertices and an even cycle decomposition of size 3 admits an $\bar{X}$-coloring or an $\hat{A}$-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 $\bar{X}$-colorable. In this short note, we discuss two algorithms for deciding whether a graph $G$ is $H$-colorable. We give a negative answer to the two questions and disprove the conjecture by finding suitable graphs, as verified by two independent algorithms.