Fast embedding of spanning trees in biased Maker–Breaker games

Given a tree T=(V,E)T=(V,E) on nn vertices, we consider the (1:q)(1:q) Maker–Breaker tree embedding game TnTn. The board of this game is the edge set of the complete graph on nn vertices. Maker wins TnTn if and only if she is able to claim all edges of a copy of TT. We prove that there exist real numbers α,ε>0α,ε>0 such that, for sufficiently large nn and for every tree TT on nn vertices with maximum degree at most nεnε, Maker has a winning strategy for the (1:q)(1:q) game TnTn, for every q≤nαq≤nα. Moreover, we prove that Maker can win this game within n+o(n)n+o(n) moves which is clearly asymptotically optimal.