Strong edge-colorings of sparse graphs with large maximum degree

A {\em strong $k$-edge-coloring} of a graph $G$ is a mapping from $E(G)$ to $\{1,2,\dots ,k\}$ such that every two adjacent edges or two edges adjacent to the same edge receive distinct colors. The {\em strong chromatic index} ${\chi }_s^{\mathrm{\prime }}(G)$ of a graph $G$ is the smallest integer $k$ such that $G$ admits a strong $k$-edge-coloring. We give bounds on ${\chi }_s^{\mathrm{\prime }}(G)$ in terms of the maximum degree $\mathrm{\Delta }(G)$ of a graph $G$. when $G$ is sparse, namely, when $G$ is $2$-degenerate or when the maximum average degree $\mathrm{M}\mathrm{a}\mathrm{d}(G)$ is small. We prove that the strong chromatic index of each $2$-degenerate graph $G$ is at most $5\mathrm{\Delta }(G)+1$. Furthermore, we show that for a graph $G$, if ${\rm Mad}(G)< 8/3$and$\Delta(G)\geq 9$,then$\chi_s’(G)\leq 3\Delta(G) -3$ (the bound $3\mathrm{\Delta }(G)-3$ is sharp) and if $\mathrm{M}\mathrm{a}\mathrm{d}(G)<3$ and $\mathrm{\Delta }(G)\ge 7$, then ${\chi }_s^{\mathrm{\prime }}(G)\le 3\mathrm{\Delta }(G)$ (the restriction $\mathrm{M}\mathrm{a}\mathrm{d}(G)<3$ is sharp).