The packing chromatic number of infinite product graphs

The packing chromatic number χρ(G) of a graph G is the smallest integer k such that the vertex set V (G) can be partitioned into disjoint classes X1, . . . , Xk, where vertices in Xi have pairwise distance greater than i. For the Cartesian product of a path and the 2-dimensional square lattice it is proved that χρ(Pm Z) = ∞ for any m ≥ 2, thus extending the result χρ(Z) = ∞ of Finbow and Rall [4]. It is also proved that χρ(Z) ≥ 10 which improves the bound χρ(Z) ≥ 9 of Goddard et al. [5]. Moreover, it is shown that χρ(G Z) < ∞ for any finite graph G. The infinite hexagonal lattice H is also considered and it is proved that χρ(H) ≤ 7 and χρ(Pm H) = ∞ for m ≥ 6.