Odd and even cycles in maker-breaker games (Q2426453)

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. This is matched by a previous result of the authors [Eur. J. Comb. 26, No. 2, 271--285 (2005; Zbl 1099.91030)]. We show that if \(q<n/2-o(n)\) then Breaker can ensure that Maker's graph is acyclic. We also consider the odd cycle game and show that for \(q<(1-1/\sqrt{2}-o(1))n\) Maker can create an odd cycle.