Odd and even cycles in Maker–Breaker games

Let Maker and Breaker alternately select respectively 1 and q previously unclaimed edges of K n until all edges have been claimed. In the even cycle game Maker’s aim is to create an even cycle. We show that if q < n 2 − o ( n ) , then Maker has a winning strategy. This is asymptotically matched by a previous result of the authors [M. Bednarska, O. Pikhurko, Biased positional games on matroids, Eur. J. Combin. 26 (2005) 271–285] that if q ≥ ⌈ n / 2 ⌉ − 1 then Breaker can ensure that Maker’s graph is acyclic. We also consider the odd cycle game and show that for q < ( 1 − 1 / 2 − o ( 1 ) ) n Maker can create an odd cycle.