Building Spanning Trees Quickly in Maker-Breaker Games

Abstract

For a tree T on n vertices, we study the Maker-Breaker game, played on the edge set of the complete graph on n vertices, which Maker wins as soon as the graph she builds contains a copy of T. We prove that if T has bounded maximum degree and $n$ is sufficiently large, then Maker can win this game within n+1 moves. Moreover, we prove that Maker can build almost every tree on n vertices in n-1 moves and provide nontrivial examples of families of trees which Maker cannot build in n-1 moves.