A Short Proof that Every Claw-Free Cubic Graph is (1,1,2,2)-Packing Colorable

TL;DR

This paper presents a shorter, simpler proof that all claw-free cubic graphs can be partitioned into two 1-packings and two 2-packings, confirming a recent coloring result.

Contribution

It offers a more concise and accessible proof of a known coloring property for claw-free cubic graphs, improving understanding and proof simplicity.

Findings

All claw-free cubic graphs are (1,1,2,2)-packing colorable

The proof simplifies previous complex arguments

Supports the conjecture with a shorter demonstration

Abstract

It was recently proved that every claw-free cubic graph admits a (1, 1, 2, 2)-packing coloring--that is, its vertex set can be partitioned into two 1-packings and two 2-packings. This result was established by Bre\v{s}ar, Kuenzel, and Rall [Discrete Mathematics 348 (8) (2025), 114477]. In this paper, we provide a simpler and shorter proof.