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) ≥ δ (and) χ(G) ≥ k, then Maker can build an odd cycle in the (1: b) game for b = O(n/log²n). We also consider the analogous game log² n 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, Łuczak 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?.