Odd and even cycles in Maker–Breaker games

AbstractLet Maker and Breaker alternately select respectively 1 and q previously unclaimed edges of Kn 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<n2−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