The Tight Bound for the Strong Chromatic Indices of Claw-Free Subcubic Graphs - Lin, Yuquan; Lin, Wensong (Graphs and combinatorics, 2023)
Let G be a graph and k a positive integer. A strong k -edge-coloring of G is a mapping ϕ : E ( G ) → { 1 , 2 , ⋯ , k } such that for any two edges e and e ′ that are either adjacent to each other or adjacent to a common edge, ϕ ( e ) ≠ ϕ ( e ′ ) . The strong chromatic index of G , denoted as χ 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 χ s ′ ( G ) ≤ 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.