Positional games on random graphs

We introduce and study Maker/Breaker-type positional games on random graphs. Our main concern is to determine the threshold probability p(F) for the existence of Maker's strategy to claim a member of F in the unbiased game played on the edges of random graph G(n,p), for various target families F of winning sets. More generally, for each probability above this threshold we study the smallest bias b such that Maker wins the (1 : b) biased game. We investigate these functions for a number of basic games, like the connectivity game, the perfect matching game, the clique game and the Hamiltonian cycle game. (C) 2005 Wiley Periodicals, Inc.