Strong edge-coloring of graphs with maximum edge weight seven

TL;DR

This paper proves an upper bound on the strong chromatic index for graphs with maximum edge weight seven and certain average degree constraints, using the discharging method.

Contribution

It establishes a new upper bound of 13 for the strong chromatic index under specified conditions and determines the maximum average degree for such graphs.

Findings

Strong upper bound of 13 for the strong chromatic index.

Applicable to graphs with maximum edge weight 7 and average degree less than 40/13.

Provides the maximum average degree for graphs with given maximum edge weight.

Abstract

A strong edge-coloring of a graph $G$ is an edge-coloring such that any two edges of distance at most two receive distinct colors. The minimum number of colors we need in order to give $G$ a strong edge-coloring is called the strong chromatic index of $G$, denoted by $\chi_s'(G)$. The maximum edge weight of $G$ is defined to be $\max\{d(u)+d(v):\ uv\in E(G)\}$. In this paper, using the discharging method, we prove that if $G$ is a graph with maximum edge weight $7$ and maximum average degree less than $\frac{40}{13}$, then $\chi_s'(G)\le 13$. Also, we determine the largest possible maximum average degree of a graph with given maximum edge weight.