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The game chromatic number Οgβ is investigated for Cartesian product Gβ‘H and corona product GβH of two graphs G and H. The exact values for the game chromatic number of Cartesian product graph of S3ββ‘Snβ is found, where Snβ is a star graph of order n+1. This extends previous results of Bartnicki et al. [1] and Sia [5] on the game chromatic number of Cartesian product graphs. Let Pmβ be the path graph on m vertices and Cnβ be the cycle graph on n vertices. We have determined the exact values for the game chromatic number of corona product graphs PmββK1β and PmββCnβ
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