The Tight Bound for the Strong Chromatic Indices of Claw-Free Subcubic Graphs

Abstract

Let G be a graph and k a positive integer. A strong k-edge-coloring of G is a mapping $$\phi : E(G)\rightarrow \{1,2,\dots ,k\}$$ such that for any two edges e and $$e'$$ that are either adjacent to each other or adjacent to a common edge, $$\phi (e)\ne \phi (e')$$ . 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 polynomial-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.