$S$-packing chromatic vertex-critical graphs

Abstract

For a non-decreasing sequence of positive integers $S = (s_1,s_2,\ldots)$, the {\em $S$-packing chromatic number} $\chi_S(G)$ of $G$ is the smallest integer $k$ such that the vertex set of $G$ can be partitioned into sets $X_i$, $i \in [k]$, where vertices in $X_i$ are pairwise at distance greater than $s_i$. In this paper we introduce $S$-packing chromatic vertex-critical graphs, $\chi_{S}$-critical for short, as the graphs in which $\chi_{S}(G-u)<\chi_{S}(G)$ for every $u\in V(G)$. This extends the earlier concept of the packing chromatic vertex-critical graphs. We show that if $G$ is $\chi_{S}$-critical, then the set $\{ \chi_{S}(G)-\chi_{S}(G-u); \, u\in V(G) \}$ can be almost arbitrary. If $G$ is $\chi_{S}$-critical and $\chi_{S}(G)=k$ ($k\in \mathbb{N}$), then $G$ is called $k$-$\chi_{S}$-critical. We characterize $3$-$\chi_{S}$-critical graphs and partially characterize $4$-$\chi_{S}$-critical graphs when $s_1>1$. We also deal with $k$-$\chi_{S}$-criticality of trees and caterpillars.