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The packing chromatic number χ ρ (G) of a graph G is the smallest integer k such that the vertex set V (G) can be partitioned into disjoint classes X 1 ,. .. , X k , with the condition that vertices in X i have pairwise distance greater than i. We show that the packing chromatic number for the hexagonal lattice H is 7. We also investigate the packing chromatic number for infinite subgraphs of the square lattice Z 2 with up to 13 rows. In particular, we establish the packing chromatic number for P 6 Z and provide new upper bounds on these numbers for the other subgraphs of interest. Finally, we explore the packing chromatic number for some infinite subgraphs of Z 2 P 2. The results are partially obtained by a computer search.

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