S-packing chromatic critical graphs

Abstract

For a non-decreasing sequence of positive integers $S=(s_1,s_2,\ldots)$, the $S$-packing chromatic number of a graph $G$ is denoted by $\chi_S(G)$. In this paper, $\chi_S$-critical graphs are introduced as the graphs $G$ such that $\chi_S(H) < \chi_S(G)$ for each proper subgraph $H$ of $G$. Several families of $\chi_S$-critical graphs are constructed, and $2$- and $3$-colorable $\chi_S$-critical graphs are presented for all packing sequences $S$, while $4$-colorable $\chi_S$-critical graphs are found for most of $S$. Cycles which are $\chi_S$-critical are characterized under different conditions. It is proved that for any graph $G$ and any edge $e \in E(G)$, the inequality $\chi_S(G - e) \ge \chi_S(G)/2$ holds. Moreover, in several important cases, this bound can be improved to $\chi_S(G - e) \ge (\chi_S(G)+1)/2$. The sharpness of the bounds is also discussed. Along the way an earlier result on $\chi_S$-vertex-critical graphs is supplemented.