Odd graph and its applications to the strong edge coloring

TL;DR

This paper investigates strong edge coloring of graphs, establishing new bounds for planar graphs and graphs with certain girth and maximum degree conditions, using properties of odd graphs and special walks.

Contribution

It introduces novel bounds on the strong chromatic index for specific classes of graphs based on girth, maximum degree, and mad, utilizing properties of odd graphs and special walks.

Findings

Planar graphs with girth at least 10Δ-4 have strong chromatic index ≤ 2Δ-1.

Graphs with girth ≥ 2Δ-1 and mad less than 2+1/(3Δ-2) have strong chromatic index ≤ 2Δ-1.

Subcubic graphs with girth ≥ 8 and mad < 2+2/23 have strong chromatic index ≤ 5.

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 $\chi_s'(G)$ 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 $\chi_s'(G) \leq 2\Delta - 1$, if $G$ is a subcubic graph with girth at least $8$ and mad$(G) < 2 + \frac{2}{23}$, then $\chi_s'(G) \leq 5$.