Strong edge-coloring of graphs with maximum edge weight seven

Abstract: A strong edge-coloring of a graph $G$ is an edge-coloring such that any two edges of distance at most two receive distinct colors. The minimum number of colors we need in order to give $G$ a strong edge-coloring is called the strong chromatic index of $G$, denoted by $\chi_s'(G)$. The maximum edge weight of $G$ is defined to be $\max{d(u)+d(v):\ uv\in E(G)}$. In this paper, using the discharging method, we prove that if $G$ is a graph with maximum edge weight $7$ and maximum average degree less than $\frac{28}{9}$, then $\chi_s'(G)\le 13$.