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arXiv · 2605.03912

Packing chromatic critical graphs with radius at most 2

Aslıhan Gür·Didem Gözüpek·Hadi Alizadeh
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Abstract

For a graph $G$ with vertex set $V(G)$ and a positive integer $i$, an $i$-packing in $G$ is a subset $X$ of $V(G)$ such that the distance between any two distinct vertices of $X$ is greater than $i$. The packing chromatic number of $G$, denoted by $\chi_{\rho}(G)$, is the smallest positive integer $k$ for which there exists a partition $X_1, X_2, \ldots, X_k$ of $V(G)$ such that $X_i$ is an $i$-packing in $G$ for every $i \in [k]$. A graph $G$ is called $\chi_\rho$-critical if $\chi_\rho(H) < \chi_\rho(G)$ holds for every proper subgraph $H$ of $G$. In this paper, we provide a structural characterization of $\chi_{\rho}$-critical graphs with radius $1$, and completely determine the $\chi_{\rho}$-critical cactus graphs with radius $2$ and diameter $2$ or $3$.

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BibTeXRIS

Aslıhan Gür, Didem Gözüpek, Hadi Alizadeh. 2026-05-05. Packing chromatic critical graphs with radius at most 2. https://arxiv.org/abs/2605.03912

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