Packing chromatic critical graphs with radius at most 2

TL;DR

This paper characterizes the structure of packing chromatic critical graphs with radius at most 2, focusing on radius 1 and specific cactus graphs with radius 2 and diameter 2 or 3.

Contribution

It provides a structural characterization of hi_rital graphs with radius 1 and fully determines those with radius 2 and diameter 2 or 3.

Findings

Characterization of hi_rital graphs with radius 1.

Complete determination of hi_rital cactus graphs with radius 2 and diameter 2 or 3.

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 $3$.