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

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 χ s ′ ( G ) , is the minimum number of colors needed to guarantee that G admits a strong edge-coloring. For any integer n ≥ 3 , let H n denote the n -prism (i.e., the Cartesian product C n □ K 2 ) and H n Δ the graph obtained from H n by replacing each vertex with a triangle. Recently, Lin and Lin (2022) asked whether χ s ′ ( H n Δ ) = 6 for any n ≥ 3 . In this short note, we answer this question in the affirmative.