Winning fast in biased Maker-Breaker games

Abstract We study the biased (1 : b ) Maker-Breaker positional games, played on the edge set of the complete graph on n vertices, K n . Given Breaker's bias b , possibly depending on n , we determine the bounds for minimal number of moves, depending on b , in which Maker can win in each of the two standard graph games, the Perfect Matching game and the Hamilton Cycle game.