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

TL;DR

This paper proves that all claw-free cubic graphs can be partitioned into four sets with specific independence and packing properties, confirming a conjecture about their packing chromatic number.

Contribution

It establishes that every claw-free cubic graph admits a (1,1,2,2)-coloring, confirming a conjecture for this class of graphs.

Findings

Every claw-free cubic graph admits a (1,1,2,2)-coloring.

Supports the conjecture that the packing chromatic number of subdivisions of subcubic graphs is at most 5.

Provides a characterization related to the Petersen graph.

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.