The packing chromatic number of infinite product graphs

The packing chromatic number chi(rho)(G) of a graph G is the smallest integer k such that the vertex set V(G) can be partitioned into disjoint classes X(l) ..... 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 chi(rho)(Pm square Z(2)) = for any m >= 2, thus extending the result chi(rho)(Z(3)) = infinity of [A. Firibow, 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 chi(rho)(Z(2)) >= 10 which improves the bound chi(rho)(Z(2)) >= 9 of [W. Goddard, S.M. Hedetnierni, 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 chi(rho)(G square Z) >= infinity for any finite graph G. The infinite hexagonal lattice H is also considered and it is proved that chi(rho)(H) >= 7 and chi(rho)(P(m)square H) = infinity for m >= 6. (C) 2008 Elsevier Ltd. All rights reserved.