The packing chromatic number of infinite product graphs

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 , where vertices in X i have pairwise distance greater than i . For the Cartesian product of a path and the two-dimensional square lattice it is proved that χ ρ ( P m □ Z 2 ) = ∞ for any m ≥ 2 , thus extending the result χ ρ ( Z 3 ) = ∞ of [A. Finbow, D.F. Rall, On the packing chromatic number of some lattices, Discrete Appl. Math. (submitted for publication) special issue LAGOS’07]. It is also proved that χ ρ ( Z 2 ) ≥ 10 which improves the bound χ ρ ( Z 2 ) ≥ 9 of [W. Goddard, S.M. Hedetniemi, S.T. Hedetniemi, J.M. Harris, D.F. Rall, Broadcast chromatic numbers of graphs, Ars Combin. 86 (2008) 33–49]. Moreover, it is shown that χ ρ ( G □ Z ) < ∞ for any finite graph G . The infinite hexagonal lattice H is also considered and it is proved that χ ρ ( H ) ≤ 7 and χ ρ ( P m □ H ) = ∞ for m ≥ 6 .