The Biased Odd Cycle Game

In this paper we consider biased Maker-Breaker games played on the edge set of a given graph G. We prove that for every δ > 0 and large enough n, there exists a constant k for which if δ(G) ≥ δn and χ(G) ≥ k, then Maker can build an odd cycle in the (1 : b) game for b = O ( n log n ) . We also consider the analogous game where Maker and Breaker claim vertices instead of edges. This is a special case of the following well known and notoriously difficult problem due to Duffus, Luczak and Rödl: is it true that for any positive constants t and b, there exists an integer k such that for every graph G, if χ(G) ≥ k, then Maker can build a graph which is not t-colorable, in the (1 : b) Maker-Breaker game played on the vertices of G?