The packing chromatic number @g"@r(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 @g"@r(P"[email protected]?Z^2)=~ for any m>=2, thus extending the result @g"@r(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 @g"@r(Z^2)>=10 which improves the bound @g"@r(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 @g"@r([email protected]?Z) =6.
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