Odd graph and its applications to the strong edge coloring

A strong edge coloring of a graph is a proper edge coloring in which every color class is an induced matching. The strong chromatic index $\chis'(G)$ of a graph $G$ is the minimum number of colors in a strong edge coloring of $G$. Let $\Delta \geq 4$ be an integer. In this note, we study the odd graphs and show the existence of some special walks. By using these results and Chang's ideas in [Discuss. Math. Graph Theory 34 (4) (2014) 723--733], we show that every planar graph with maximum degree at most $\Delta$ and girth at least $10 \Delta - 4$ has a strong edge coloring with $2\Delta - 1$ colors. In addition, we prove that if $G$ is a graph with girth at least $2\Delta - 1$ and mad$(G) < 2 + \frac{1}{3\Delta - 2}$, where $\Delta$ is the maximum degree and $\Delta \geq 4$, then $\chis'(G) \leq 2\Delta - 1$, if $G$ is a subcubic graph with girth at least $8$ and mad$(G) < 2 + \frac{2}{23}$, then $\chi_s'(G) \leq 5$.