The packing chromatic number of the infinite square lattice is between 13 and 15
An i-packing in a graph G is a set of vertices at pairwise distance greater than i. For a nondecreasing sequence of integers S = (s1, s2, . . .), the S-packing chromatic number of a graph G is the least integer k such that there exists a coloring of G into k colors where each set of vertices colored i, i = 1, . . . , k, is an si-packing. This paper describes various subdivisions of an i-packing into j-packings (j > i) for the hexagonal, square and triangular lattices. These results allow us to bound the Spacking chromatic number for these graphs, with more precise bounds and exact values for sequences S = (si, i ∈ N ∗), si = d+ ⌊(i− 1)/n⌋.