Strong edge coloring of graphs with maximum degree $6$

Let $G$ be a graph. Under a strong edge coloring of $G$, every color class is an induced matching. The strong chromatic index of $G$, denoted by ${\chi }_s^{\mathrm{\prime }}(G)$, is the smallest integer $k$ such that $G$ admits a strong edge coloring with $k$ colors. Denote by $\mathrm{\Delta }(G)$ the maximum degree of $G$. In this paper, we prove that every graph $G$ with $\mathrm{\Delta }(G)\le 6$ satisfies ${\chi }_s^{\mathrm{\prime }}(G)\le 57$, improving the best known upper bound $60$.