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 PP we seek a graph (board of the game) with the smallest number of edges on which Maker can build a subgraph that satisfies PP. 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 nn vertices and about 10n/710n/7 edges, on the other hand, at least 11n/811n/8 edges are required; (2) for the spanning kk-connectivity game, there is a winning board with nn vertices and (1+ok(1))kn(1+ok(1))kn edges; (3) for the Hamiltonicity game, there is a winning board of constant average degree; (4) for a tree TT on nn vertices of bounded maximum degree ΔΔ, there is a graph GG on nn vertices and at most f(Δ)⋅nf(Δ)⋅n edges, on which Maker can construct a copy of TT. We also discuss biased versions of these games and argue that the picture changes quite drastically there.