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Very recently, a long-standing open question proposed by Bodlaender in 1991was answered: the graph coloring game is PSPACE-complete. In 2019, Andres andLock proposed five variants of the graph coloring game and left open thequestion of PSPACE-hardness related to them. In this paper, we prove that thesevariants are PSPACE-complete for the graph coloring game and also for thegreedy coloring game, even if the number of colors is the chromatic number.Finally, we also prove that a connected version of the graph coloring game,proposed by Charpentier et al. in 2019, is PSPACE-complete.

Hardness of some variants of the graph coloring game