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 $χ_ρ(G)$, 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 $χ_ρ$-critical if $χ_ρ(H) < χ_ρ(G)$ holds for every proper subgraph $H$ of $G$. In this paper, we provide a structural characterization of $χ_ρ$-critical graphs with radius $1$, and completely determine the $χ_ρ$-critical cactus graphs with radius $2$ and diameter $2$ or $3$.

翻译：对于图$G$，设$V(G)$为其顶点集，$i$为正整数。$G$中的$i$-填装是指顶点子集$X \subseteq V(G)$，使得$X$中任意两个不同顶点之间的距离大于$i$。图$G$的填色色数，记作$χ_ρ(G)$，是满足存在顶点集$V(G)$的一个划分$X_1, X_2, \ldots, X_k$（其中对每个$i \in [k]$而言，$X_i$都是$G$中的$i$-填装）的最小正整数$k$。若对$G$的每个真子图$H$均有$χ_ρ(H) < χ_ρ(G)$，则称图$G$是$χ_ρ$-临界的。本文给出了半径为$1$的$χ_ρ$-临界图的结构刻画，并完全确定了半径为$2$且直径为$2$或$3$的$χ_ρ$-临界仙人掌图。