Game chromatic number of Cartesian product graphs

The game chromatic number χ g is considered for the Cartesian product G □ H of two graphs G and H. We determine exact values of χ g ( G □ H ) when G and H belong to certain classes of graphs, and show that, in general, the game chromatic number χ g ( 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 col g ( G □ H ) of the Cartesian product.