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 ${\chi }_s^{\prime }(G)$. 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 $\frac{40}{13}$, then ${\chi }_s^{\prime }(G)\le 13$. Also, we determine the largest possible maximum average degree of a graph with given maximum edge weight.

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