(d,n)-packing colorings of infinite lattices

For a nondecreasing sequence of integers S=(s1,s2,…) an S-packing k-coloring of a graph G is a mapping from V(G) to {1,2,…,k} such that vertices with color i∈{1,2,…,k} have pairwise distance greater than si. A natural restriction of this concept obtained by setting si=d+⌊i−1n⌋ is called 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. We study (d,n)-packing chromatic colorings of several lattices including the infinite square, hexagonal, triangular, eight-regular, octagonal and two-row square lattice.