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 10 n / 7 edges, on the other hand, at least 11 n / 8 edges are required; (2) for the spanning k -connectivity game, there is a winning board with n vertices and ( 1 + o k ( 1 ) ) k n 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 Δ , there is a graph G on n vertices and at most f ( Δ ) ⋅ 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.