On the packing coloring of base-3 Sierpi\'{n}ski and $H$ graphs

Abstract

For a nondecreasing sequence of integers $S=(s_1, s_2, \ldots)$ an $S$-packing $k$-coloring of a graph $G$ is a mapping from $V(G)$ to $\{1, 2,\ldots,k\}$ such that vertices with color $i$ have pairwise distance greater than $s_i$. By setting $s_i = d + \lfloor \frac{i-1}{n} \rfloor$ we obtain a $(d,n)$-packing coloring of a graph $G$. The smallest integer $k$ for which there exists a $(d,n)$-packing coloring of $G$ is called the $(d,n)$-packing chromatic number of $G$. In the special case when $d$ and $n$ are both equal to one we speak of the packing chromatic number of $G$. We determine the packing chromatic number of the base-3 Sierpi\'{n}ski graphs $S_k$ and provide new results on $(d,n)$-packing chromatic colorings, $d \le 2$, for this class of graphs. By using a dynamic algorithm, we establish the packing chromatic number for $H$-graphs $H(r)$.