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For a graph G, a monotone increasing graph property P and positive integer q,we define the Client-Waiter game to be a two-player game which runs as follows.In each turn Waiter is offering Client a subset of at least one and at most q+1unclaimed edges of G from which Client claims one, and the rest are claimed byWaiter. The game ends when all the edges have been claimed. If Client's graphhas property P by the end of the game, then he wins the game, otherwise Waiteris the winner. In this paper we study several Client-Waiter games on the edgeset of the complete graph, and the H-game on the edge set of the random graph.For the complete graph we consider games where Client tries to build a largestar, a long path and a large connected component. We obtain lower and upperbounds on the critical bias for these games and compare them with thecorresponding Waiter-Client games and with the probabilistic intuition. For theH-game on the random graph we show that the known results for the correspondingMaker-Breaker game are essentially the same for the Client-Waiter game, and weextend those results for the biased games and for trees.

Client-Waiter games on complete and random graphs