A note on S-packing colorings of lattices

Let a1,a2,…,ak be positive integers. An (a1,a2,…,ak)-packing coloring of a graph G is a mapping from V(G) to {1,2,…,k} such that vertices with color i have pairwise distance greater than ai. In this paper, we study (a1,a2,…,ak)-packing colorings of several lattices including the infinite square, triangular, and hexagonal lattices. For k small, we determine all ai such that these graphs have packing colorings. We also give some exact values and asymptotic bounds.