Strong chromatic index of k-degenerate graphs

TL;DR

This paper improves upper bounds on the strong chromatic index of k-degenerate graphs, especially for 2-degenerate graphs and certain subclasses, providing tighter results than previous studies.

Contribution

It presents new upper bounds for the strong chromatic index of k-degenerate graphs, including specific improvements for 2-degenerate graphs and subclasses where all 3+-vertices induce a forest.

Findings

Strong chromatic index of k-degenerate graphs is at most (4k-2)Δ(G) - 2k^2 + 1.

For 2-degenerate graphs, the bound is improved to 6Δ(G) - 7.

Special subclass graphs have an even tighter bound of 4Δ(G) - 3.

Abstract

A {\em strong edge coloring} of a graph $G$ is a proper edge coloring in which every color class is an induced matching. The {\em strong chromatic index} $\chiup_{s}'(G)$ of a graph $G$ is the minimum number of colors in a strong edge coloring of $G$. In this note, we improve a result by D{\k e}bski \etal [Strong chromatic index of sparse graphs, arXiv:1301.1992v1] and show that the strong chromatic index of a $k$-degenerate graph $G$ is at most $(4k-2) \cdot \Delta(G) - 2k^{2} + 1$. As a direct consequence, the strong chromatic index of a $2$-degenerate graph $G$ is at most $6\Delta(G) - 7$, which improves the upper bound $10\Delta(G) - 10$ by Chang and Narayanan [Strong chromatic index of 2-degenerate graphs, J. Graph Theory 73 (2013) (2) 119--126]. For a special subclass of $2$-degenerate graphs, we obtain a better upper bound, namely if $G$ is a graph such that all of its $3^{+}$-vertices induce a forest, then $\chiup_{s}'(G) \leq 4 \Delta(G) -3$; as a corollary, every minimally $2$-connected graph $G$ has strong chromatic index at most $4 \Delta(G) - 3$. Moreover, all the results in this note are best possible in some sense.