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