A note on strong edge-coloring of claw-free cubic graphs

Abstract

A strong edge-coloring of a graph G is an edge-coloring of G such that any two edges that are either adjacent to each other or adjacent to a common edge receive distinct colors. The strong chromatic index of G, denoted by $\chi '_s(G)$, is the minimum number of colors needed to guarantee that G admits a strong edge-coloring. For any integer $n\ge 3$, let $H_n$ denote the n-prism (i.e., the Cartesian product $C_n\square K_2$) and $H_n^{\Delta }$ the graph obtained from $H_n$ by replacing each vertex with a triangle. Recently, Lin and Lin (2022) asked whether $\chi '_s(H_n^{\Delta })=6$ for any $n\ge 3$. In this short note, we answer this question in the affirmative.