Fast embedding of spanning trees in biased Maker-Breaker games

Abstract

Given a tree $T=(V,E)$ on $n$ vertices, we consider the $(1:q)$ Maker-Breaker tree embedding game ${\mathcal{T}}_n$. The board of this game is the edge set of the complete graph on $n$ vertices. Maker wins ${\mathcal{T}}_n$ if and only if he is able to claim all edges of a copy of $T$. We prove that there exist real numbers $\alpha ,\epsilon >0$ such that, for sufficiently large $n$ and for every tree $T$ on $n$ vertices with maximum degree at most $n^{\epsilon }$, Maker has a winning strategy for the $(1:q)$ game ${\mathcal{T}}_n$, for every $q\le n^{\alpha }$. Moreover, we prove that Maker can win this game within $n+o(n)$ moves which is clearly asymptotically optimal.

20 pages