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arXiv · 2409.15455

Claw-free cubic graphs are $(1, 1, 2, 2)$-colorable

Boštjan Brešar·Kirsti Kuenzel·Douglas F. Rall
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Abstract

A $(1,1,2,2)$-coloring of a graph is a partition of its vertex set into four sets two of which are independent and the other two are $2$-packings. In this paper, we prove that every claw-free cubic graph admits a $(1,1,2,2)$-coloring. This implies that the conjecture from [Packing chromatic number, $(1,1,2,2)$-colorings, and characterizing the Petersen graph, Aequationes Math.\ 91 (2017) 169--184] that the packing chromatic number of subdivisions of subcubic graphs is at most $5$ is true in the case of claw-free cubic graphs.

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Boštjan Brešar, Kirsti Kuenzel, Douglas F. Rall. 2024-09-23. Claw-free cubic graphs are $(1, 1, 2, 2)$-colorable. https://arxiv.org/abs/2409.15455

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