Asymptotic thresholds for (a : b) minimum-degree games

We investigate the (a:b) Maker-Breaker subgraph game played on the edge set of the complete graph Kn, where n,a,b∈N, and Maker's objective is to construct a member of a prescribed family of graphs H, while Breaker aims to prevent this. In our study, Breaker moves first, and H is taken to be either the family of connected spanning subgraphs or the family of spanning subgraphs with minimum-degree at least k=k(n). For the (a:b) minimum-degree-k game, we determine the asymptotically optimal threshold bias across a wide range of values for a. For the (a:b) connectivity game, we resolve an open problem posed by Hefetz et al. (2012) by identifying the exact leading term in the asymptotic behavior of the threshold bias when a=clnn.