Rich-neighbor 5-edge-colorings of subcubic graphs whose 3-vertices induce a matching

Let $\phi$ be a proper edge-coloring of a graph $G$. An edge is called rich if all edges adjacent to it receive pairwise distinct colors. The coloring $\phi$ is a rich-neighbor edge-coloring if every non-isolated edge is adjacent to a rich edge. Petruševski and Škrekovski introduced this notion as a relaxation of strong edge-coloring and conjectured that every connected subcubic graph other than $K_4$ admits a rich-neighbor $5$-edge-coloring. We establish the conjectured bound for every subcubic graph $G$ satisfying $\mathrm{\Delta }(G[V_3])\le 1,$ where $V_3$ is the set of vertices of degree $3$ in $G$. Equivalently, the $3$-vertices of $G$ induce a matching together with isolated vertices. This extends the result of Chen, Tan and Zhou for subcubic graphs with no adjacent $3$-vertices.