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

Maidoun Mortada, Ayman El Zein

TL;DR

This work addresses the packing-coloring problem for claw-free cubic graphs by proving, via a concise purely combinatorial argument, that two disjoint $2$-packings can be chosen to destroy all triangles in any cubic graph, and then exploiting an odd-cycle reduction to obtain a $(1,1,2,2)$-packing coloring for claw-free instances. The key technical contribution is the Lemma guaranteeing a triangle-free remainder after removing a maximal packing pair, coupled with an explicit reduction mechanism that drives the remaining odd cycles to zero in the claw-free case. As a result, the authors deliver a shorter and more transparent proof of the known result by Brešar, Kuenzel, and Rall, while also highlighting a general tool (the odd-cycle reduction) with potential applicability to broader packing-coloring questions. The work also opens avenues for stronger colorings and poses natural open problems about relaxing claw-freeness and extending the coloring spectrum to $(1,1,2,3)$.

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 Brešar, Kuenzel, and Rall [Discrete Mathematics 348 (8) (2025), 114477]. In this paper, we provide a simpler and shorter proof.

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

TL;DR

This work addresses the packing-coloring problem for claw-free cubic graphs by proving, via a concise purely combinatorial argument, that two disjoint -packings can be chosen to destroy all triangles in any cubic graph, and then exploiting an odd-cycle reduction to obtain a -packing coloring for claw-free instances. The key technical contribution is the Lemma guaranteeing a triangle-free remainder after removing a maximal packing pair, coupled with an explicit reduction mechanism that drives the remaining odd cycles to zero in the claw-free case. As a result, the authors deliver a shorter and more transparent proof of the known result by Brešar, Kuenzel, and Rall, while also highlighting a general tool (the odd-cycle reduction) with potential applicability to broader packing-coloring questions. The work also opens avenues for stronger colorings and poses natural open problems about relaxing claw-freeness and extending the coloring spectrum to .

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 Brešar, Kuenzel, and Rall [Discrete Mathematics 348 (8) (2025), 114477]. In this paper, we provide a simpler and shorter proof.
Paper Structure (4 sections, 2 theorems, 1 equation)

This paper contains 4 sections, 2 theorems, 1 equation.

Table of Contents

  1. Introduction
  2. The Key Lemma: Breaking All Triangles with Two 2-Packings
  3. New Proof
  4. Open Problem

Key Result

Lemma 2.1

Let $G$ be a cubic graph. Then there exist two disjoint 2-packings in $G$, say $X$ and $Y$, such that $G[V(G)\setminus (X\cup Y)]$ is triangle-free.

Theorems & Definitions (4)

  • Lemma 2.1
  • proof
  • Theorem 3.1
  • proof