Strong edge coloring of graphs with maximum degree $6$

Abstract

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 $\chi'_s(G)$, is the smallest integer $k$ such that $G$ admits a strong edge coloring with $k$ colors. Denote by $\Delta(G)$ the maximum degree of $G$. In this paper, we prove that every graph $G$ with $\Delta(G)\le 6$ satisfies $\chi'_s(G)\le 57$, improving the best known upper bound $60$.