The tight bound for the strong chromatic indices of claw-free subcubic graphs

TL;DR

This paper proves that the strong chromatic index of claw-free subcubic graphs (excluding the triangular prism) is at most 7, improving the previous bound of 8, and provides a linear-time coloring algorithm.

Contribution

It establishes a tighter upper bound of 7 for the strong chromatic index of claw-free subcubic graphs and offers an efficient coloring algorithm.

Findings

Proved the upper bound of 7 for the strong chromatic index.

Developed a linear-time algorithm for strong 7-edge-colorings.

Constructed infinite graphs with chromatic index exactly 7.

Abstract

Let $G$ be a graph and $k$ a positive integer. A strong $k$-edge-coloring of $G$ is a mapping $\phi: E(G)\to \{1,2,\dots,k\}$ such that for any two edges $e$ and $e'$ that are either adjacent to each other or adjacent to a common edge, $\phi(e)\neq \phi(e')$. The strong chromatic index of $G$, denoted as $\chi'_{s}(G)$, is the minimum integer $k$ such that $G$ has a strong $k$-edge-coloring. Lv, Li and Zhang [Graphs and Combinatorics 38 (3) (2022) 63] proved that if $G$ is a claw-free subcubic graph other than the triangular prism then $\chi_s'(G)\le 8$. In addition, they asked if the upper bound $8$ can be improved to $7$. In this paper, we answer this question in the affirmative. Our proof implies a linear-time algorithm for finding strong $7$-edge-colorings of such graphs. We also construct infinitely many claw-free subcubic graphs with their strong chromatic indices attaining the bound $7$.