The biased odd cycle game

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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/² 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, 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?