Strong edge-coloring of sparse graphs with Ore-degree 7 or 8

TL;DR

This paper proves new bounds on the strong chromatic index for sparse graphs with Ore-degree 7 or 8, improving previous bounds by applying discharging and Hall's marriage theorem.

Contribution

It establishes improved upper bounds on the strong chromatic index for graphs with specific Ore-degree and average degree constraints, advancing the understanding of strong edge-coloring.

Findings

For Ore-degree 7, strong chromatic index at most 13 under certain conditions.

For Ore-degree 8, strong chromatic index at most 20 under certain conditions.

Improves previous bounds from 40/13 to 34/11 for Ore-degree 7.

Abstract

In a strong edge-coloring of a graph $G=(V,E)$, any two edges of distance at most $2$ get distinct colors. The strong chromatic index of $G$, denoted by $\chi_s'(G)$, is the minimum number of colors needed in a strong edge-coloring of $G$. The Ore-degree of $G$ is defined by $\max\{d(u)+d(v):uv\in E\}$. In this paper, we apply the discharging method and make use of Hall's marriage theorem to prove two results toward a conjecture by Chen et al. First, we prove that if $G$ is a graph with Ore-degree $7$ and maximum average degree less than $\frac{34}{11}$, then $\chi_s'(G)\le 13$. This result improves the previous best bound from $\frac{40}{13}$ to $\frac{34}{11}$. Second, we prove that if $G$ is a graph with Ore-degree $8$ and maximum average degree less than $\frac{113}{31}$, then $\chi_s'(G)\le 20$.