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(log2nn). 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, {\L}uczak
and R\"{o}dl: 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?Comment: 10 page