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In this paper we investigate the game chromatic number for completemultipartite graphs. We devise several strategies for Alice, and one strategyfor Bob, and we prove their optimality in all complete multipartite graphs withno singletons. All the strategies presented are computable in linear time, andthe values of the game chromatic number depend directly only on the number andthe sizes of sets in the partition.

The Game Chromatic Number of Complete Multipartite Graphs with No Singletons