On the precise value of the strong chromatic-index of a planar graph with a large girth

Abstract

A strong $k$-edge-coloring of a graph $G$ is a mapping from $E(G)$ to $\{1,2,\ldots,k\}$ such that every pair of distinct edges at distance at most two receive different colors. The strong chromatic index $\chi'_s(G)$ of a graph $G$ is the minimum $k$ for which $G$ has a strong $k$-edge-coloring. Denote $\sigma(G)=\max_{xy\in E(G)}\{\operatorname{deg}(x)+\operatorname{deg}(y)-1\}$. It is easy to see that $\sigma(G) \le \chi'_s(G)$ for any graph $G$, and the equality holds when $G$ is a tree. For a planar graph $G$ of maximum degree $\Delta$, it was proved that $\chi'_s(G) \le 4 \Delta +4$ by using the Four Color Theorem. The upper bound was then reduced to $4\Delta$, $3\Delta+5$, $3\Delta+1$, $3\Delta$, $2\Delta-1$ under different conditions for $\Delta$ and the girth. In this paper, we prove that if the girth of a planar graph $G$ is large enough and $\sigma(G)\geq \Delta(G)+2$, then the strong chromatic index of $G$ is precisely $\sigma(G)$. This result reflects the intuition that a planar graph with a large girth locally looks like a tree.