On the packing chromatic number of some lattices

AbstractFor a positive integer k, a k-packing in a graph G is a subset A of vertices such that the distance between any two distinct vertices from A is more than k. The packing chromatic number of G is the smallest integer m such that the vertex set of G can be partitioned as V1,V2,…,Vm where Vi is an i-packing for each i. It is proved that the planar triangular lattice T and the three-dimensional integer lattice Z3 do not have finite packing chromatic numbers