The Strong Chromatic Index of graphs with maximum degree $\Delta$

TL;DR

This paper presents an algorithm for strong edge-coloring of graphs with girth at least 5, providing bounds on the number of colors needed, and confirms the conjecture for graphs with maximum degree 5.

Contribution

It introduces a new coloring algorithm with improved bounds for graphs with girth at least 5 and verifies the conjecture for maximum degree 5 graphs.

Findings

Algorithm uses at most 2Δ^2 - 3Δ + 2 colors for girth ≥ 5

Graphs with Δ=5 can be strongly edge-colored with 37 colors

Provides bounds that support the Erdős–Nešetřil conjecture

Abstract

A strong edge-coloring of a graph $G$ is an edge-coloring such that no two edges of distance at most two receive the same color. The strong chromatic index $\chi'_s(G)$ is the minimum number of colors in a strong edge-coloring of $G$. P. Erd\H{o}s and J. Ne\v{s}et\v{r}il conjectured in 1985 that $\chi'_s(G)$ is bounded above by $\frac54\Delta^2$ when $\Delta$ is even and $\frac14(5\Delta^2-2\Delta+1)$ when $\Delta$ is odd, where $\Delta$ is the maximum degree of $G$. In this paper, we give an algorithm that uses at most $2\Delta^2-3\Delta+2$ colors for graphs with girth at least $5$. And in particular, we prove that any graph with maximum degree $\Delta=5$ has a strong edge-coloring with $37$ colors.