🤖 AI Summary
This study investigates the structure of packing chromatic critical graphs with radius at most two. By partitioning the vertex set into i-packing subsets—subsets in which any two vertices are at distance greater than i—and leveraging properties related to graph radius, diameter, and the structural characteristics of cactus graphs, the authors provide the first complete characterization of all packing chromatic critical graphs of radius one. Furthermore, they fully determine the class of packing chromatic critical cactus graphs of radius two whose diameter is either two or three. This work offers a systematic classification of critical graph structures under specified constraints on radius and diameter, significantly advancing the theoretical understanding of packing chromatic criticality.
📝 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 $χ_ρ(G)$, 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 $3$.