Odd graph and its applications to the strong edge coloring

Abstract

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 ${\chi }_s^{\prime }(G)$ of a graph $G$ is the minimum number of colors in a strong edge coloring of $G$. Let $\Delta \ge 4$ 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 $\Delta \ge 4$, then ${\chi }_s^{\prime }(G)\le 2\Delta -1$, if $G$ is a subcubic graph with girth at least $8$ and mad$(G)<2+\frac{2}{23}$, then ${\chi }_s^{\prime }(G)\le 5$.

7 pages

References in corpus (1)

• Strong edge-colorings for k-degenerate graphs