A heuristic approach for searching [formula omitted]-packing colorings of infinite lattices

Abstract

A (d,n)-packing k-coloring of a graph G for integers d and n is a mapping from V(G) to the set {1,2,…,k} such that vertices with color i∈{1,2,…,k} have pairwise distance greater than d+⌊i−1n⌋. The smallest integer k for which there exists a (d,n)-packing coloring of a graph G is called the (d,n)-packing chromatic number of G. We propose a new heuristic approach to find (d,n)-packing colorings of infinite lattices. The proposed algorithms have been able to obtain several (d,n)-packing colorings of the infinite square, hexagonal, triangular, eight-regular and octagonal lattice. The obtained results improve upper bounds on the corresponding (d,n)-packing chromatic numbers for these graphs.