Waiter-Client and Client-Waiter planarity, colorability and minor games

For a finite set X , a family of sets F ? 2 X and a positive integer q , we consider two types of two player, perfect information games with no chance moves. In each round of the ( 1 : q ) Waiter-Client game ( X , F ) , the first player, called Waiter, offers the second player, called Client, q + 1 elements of the board X which have not been offered previously. Client then chooses one of these elements which he claims and the remaining q elements are claimed by Waiter. Waiter wins this game if by the time every element of X has been claimed by some player, Client has claimed all elements of some A ? F ; otherwise Client is the winner. Client-Waiter games are defined analogously, the main difference being that Client wins the game if he manages to claim all elements of some A ? F and Waiter wins otherwise. In this paper we study the Waiter-Client and Client-Waiter versions of the non-planarity, K t -minor and non- k -colorability games. For each such game, we give a fairly precise estimate of the unique integer q at which the outcome of the game changes from Client's win to Waiter's win. We also discuss the relation between our results, random graphs, and the corresponding Maker-Breaker and Avoider-Enforcer games.

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