Strong edge-coloring of 2-degenerate graphs

TL;DR

This paper improves the upper bound on the strong chromatic index for 2-degenerate graphs, showing it is less than 5 times the maximum degree minus a sublinear term, for large degrees.

Contribution

It provides a tighter upper bound on the strong chromatic index of 2-degenerate graphs, refining previous results by incorporating a sublinear correction term.

Findings

Improved upper bound: $oxed{ ext{strong chromatic index} \, \leq 5\Delta - \Delta^{1/2-\epsilon} + 2}$ for large $\, \Delta$.

Applicable for any $0<\epsilon<1/2$, with the bound valid when $\Delta > 4^{1/(2\epsilon)}$.

Enhances understanding of edge-coloring complexity in 2-degenerate graphs.

Abstract

A strong edge-coloring of a graph $G$ is an edge-coloring in which every color class is an induced matching, and the strong chromatic index $\chi_s'(G)$ is the minimum number of colors needed in strong edge-colorings of $G$. A graph is $2$-degenerate if every subgraph has minimum degree at most $2$. Choi, Kim, Kostochka, and Raspaud (2016) showed $\chi_s'(G) \leq 5\Delta +1$ if $G$ is a $2$-degenerate graph with maximum degree $\Delta$. In this article, we improve it to $\chi_s'(G)\le 5\Delta-\Delta^{1/2-\epsilon}+2$ when $\Delta>4^{1/(2\epsilon)}$ for any $0<\epsilon<1/2$.