  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.
