On the rich-neighbor edge-coloring of sparse graphs

Let $\varphi$ be a proper edge-coloring of a graph $G$. An edge $e$ is rich if the edges adjacent to $e$ receive distinct colors. In particular, a pendant edge and an isolated edge are both rich. Petruševski and Škrekovski (2024) introduced the concept of rich-neighbor edge-coloring as a weakening of strong edge-coloring. A proper $k$-edge-coloring $\varphi$ is a rich-neighbor $k$-coloring if each non-isolated edge is adjacent to at least one rich edge. Petruševski and Škrekovski (2024) conjectured that every connected subcubic graph admits a rich-neighbor 5-coloring except for $K_4$. We show that if $G$ is a subcubic graph, then

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  [(1)] if $mad\left(G\right)<\frac{36}{13}$, then $G$ has a rich-neighbor 6-coloring; and
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  [(2)] if $G$ is claw-free, then $G$ has a rich-neighbor 6-coloring.