  We consider the graph coloring game, a game in which two players take turns
properly coloring the vertices of a graph, with one player attempting to
complete a proper coloring, and the other player attempting to prevent a proper
coloring. We show that if a graph $G$ has a proper coloring in which the game
coloring number of each bicolored subgraph is bounded, then the game chromatic
number of $G$ is bounded. As a corollary to this result, we show that for two
graphs $G_1$ and $G_2$ with bounded game coloring number, the Cartesian product
$G_1 \square G_2$ has bounded game chromatic number, answering a question of X.
Zhu. We also obtain an upper bound on the game chromatic number of the strong
product $G_1 \boxtimes G_2$ of two graphs.
