On the packing chromatic number of some lattices

For a positive integer kk, a kk-packing in a graph GG is a subset AA of vertices such that the distance between any two distinct vertices from AA is more than kk. The packing chromatic number of GG is the smallest integer mm such that the vertex set of GG can be partitioned as V1,V2,…,VmV1,V2,…,Vm where ViVi is an ii-packing for each ii. It is proved that the planar triangular lattice TT and the three-dimensional integer lattice Z3Z3 do not have finite packing chromatic numbers.