S-packing chromatic critical paths and cycles

Abstract

Let $S=(s_1,s_2,\ldots)$ be a non-decreasing sequence of positive integers. For a graph $G$ with vertex set $V(G)$, a labeling $\phi \colon V(G)\to \{1,\ldots,k\}$ is an $S$-packing $k$-coloring if, whenever two distinct vertices $u,v\in V(G)$ are assigned the same color $i$, their distance in $G$ is greater than $s_i$. The minimum $k$ for which $G$ admits such a coloring is the $S$-packing chromatic number of $G$. A graph $G$ is $\chi_S$-vertex-critical if $\chi_S(G-v) < \chi_S(G)$ for every $v \in V(G)$, and it is $\chi_S$-critical if $\chi_S(H) < \chi_S(G)$ holds for every proper subgraph $H$ of $G$. In this paper, the exact value of $\chi_S(P_n)$ is determined for every path of order $n$ and for every packing sequence $S$ where $s_i < 2^i$ holds for each entry $s_i$. As a consequence, $\chi_S$-critical and $\chi_S$-vertex-critical paths are identified for each such sequence $S$. In addition, we extend earlier results on $\chi_S$-critical cycles and provide a complete characterization of $\chi_S$-critical and $\chi_S$-vertex-critical cycles for packing sequences $S= (1, s_2, \dots )$ with $s_2 \in \{2,3\}$ and $s_3,s_4 \in \{4,5,6,7\}$.