Strong chromatic index of sparse graphs

TL;DR

This paper establishes new upper bounds for the strong chromatic index of sparse graphs, including k-degenerate and chordless graphs, confirming a recent conjecture and improving previous bounds.

Contribution

It proves a new upper bound for the strong chromatic index of k-degenerate graphs and improves bounds for chordless graphs, confirming a recent conjecture.

Findings

Strong chromatic index bound for k-degenerate graphs: (4k-1)Δ(G)-k(2k+1)+1

Improved upper bound for chordless graphs: 4Δ - 3

Bounds apply to list strong edge coloring as well

Abstract

A coloring of the edges of a graph $G$ is strong if each color class is an induced matching of $G$. The strong chromatic index of $G$, denoted by $\chi_{s}^{\prime}(G)$, is the least number of colors in a strong edge coloring of $G$. In this note we prove that $\chi_{s}^{\prime}(G)\leq (4k-1)\Delta (G)-k(2k+1)+1$ for every $k$-degenerate graph $G$. This confirms the strong version of conjecture stated recently by Chang and Narayanan [3]. Our approach allows also to improve the upper bound from [3] for chordless graphs. We get that $% \chi_{s}^{\prime}(G)\leq 4\Delta -3$ for any chordless graph $G$. Both bounds remain valid for the list version of the strong edge coloring of these graphs.