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+o"k(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 @D, there is a graph G on n vertices and at most f(@D)@?n edges, on which Maker can construct a copy of T. We also discuss biased versions of these games and argue that the picture changes quite drastically there.