On the packing coloring of base-3 Sierpiński graphs 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 obtain the packing chromatic number of G. We determine the packing chromatic number of base-3 Sierpinski graphs and provide new results on (d, n)-packing chromatic colorings for this class of graphs. By using a dynamic algorithm, we establish the packing chromatic number for H-graphs.