Game chromatic number of Cartesian product graphs

The game chromatic number χg is considered for the Cartesian product G□H of two graphs G and H. We determine exact values of χg(G□H) when G and H belong to certain classes of graphs, and show that, in general, the game chromatic number χg(G□H) is not bounded from above by a function of game chromatic numbers of graphs G and H. An analogous result is proved for the game coloring number colg(G□H) of the Cartesian product.