Strong edge-coloring of 2-degenerate graphs

A strong edge-coloring of a graph $G$ is an edge-coloring in which every color class is an induced matching, and the strong chromatic index ${\chi }_s^{\mathrm{\prime }}(G)$ is the minimum number of colors needed in strong edge-colorings of $G$. A graph is $2$-degenerate if every subgraph has minimum degree at most $2$. Choi, Kim, Kostochka, and Raspaud (2016) showed ${\chi }_s^{\mathrm{\prime }}(G)\le 5\mathrm{\Delta }+1$ if $G$ is a $2$-degenerate graph with maximum degree $\mathrm{\Delta }$. In this article, we improve it to ${\chi }_s^{\mathrm{\prime }}(G)\le 5\mathrm{\Delta }-{\mathrm{\Delta }}^{1\mathrm{/}2-\epsilon }+2$ when $\mathrm{\Delta }>4^{1\mathrm{/}(2\epsilon )}$ for any $0<\epsilon <1\mathrm{/}2$.