Global Maker-Breaker games on sparse graphs

In this paper we consider Maker-Breaker games, played on the edges of sparse graphs. For a given graph property P we seek a graph (board of the game) with the smallest number of edges on which Maker can build a subgraph that satisfies P. In this paper we focus on global properties. We prove the following results: 1) for the positive minimum degree game, there is a winning board with n vertices and about 10n/7 edges, on the other hand, at least 11n/8 edges are required; 2) for the spanning k-connectivity game, there is a winning board with n vertices and (1+ok(1))kn edges; 3) for the Hamiltonicity game, there is a winning board of constant average degree; 4) for a tree T on n vertices of bounded maximum degree ∆, there is a graph G on n vertices and at most f(∆) · n edges, on which Maker can construct a copy of T . We also discuss biased versions on these games and argue that the picture changes quite drastically there.