Subdivision into i-packings and S-packing chromatic number of some lattices

Abstract

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 S-packing chromatic number for these graphs, with more precise bounds and exact values for sequences S = (si; i ϵ N), si = d + b(i - 1)/n.