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

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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.

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Acknowledgements

We are very grateful to the reviewers for carefully reading this paper and putting forward many detailed and valuable suggestions to improve the paper.

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This work was supported by NSFC under grant 11771080.

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  1. School of Mathematics, Southeast University, Nanjing, 210096, People’s Republic of China

    Yuquan Lin & Wensong Lin

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  1. Yuquan Lin
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Lin, Y., Lin, W. The Tight Bound for the Strong Chromatic Indices of Claw-Free Subcubic Graphs. Graphs and Combinatorics 39, 58 (2023). https://doi.org/10.1007/s00373-023-02655-7

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