On the threshold for the Maker-Breaker H-game

We study the Maker-Breaker H-game played on the edge set of the random graph G(n, p). In this game, two players, Maker and Breaker, alternately claim unclaimed edges of G(n, p), until all the edges are claimed. Maker wins if he claims all the edges of a graph H; Breaker wins otherwise. Recently, Müller and Stojaković determined the threshold for the graph property that Maker can win this game if H is a k-clique. Extending their result, we determine the threshold for a large class of graphs, namely those which contain a cycle and whose 2-density is not determined by a K3 subgraph. In particular, we prove that for every such graph H the threshold coincides with Θ(1/bH), where bH is the threshold bias. This is joint work with Miloš Stojaković and Angelika Steger.