Fast embedding of spanning trees in biased Maker-Breaker games

TL;DR

This paper proves that Maker can quickly embed any large bounded-degree tree in a biased Maker-Breaker game on a complete graph, with winning strategies that are nearly optimal in move count.

Contribution

It establishes a fast, universal embedding strategy for Maker in biased games for large trees with bounded maximum degree, extending previous results.

Findings

Maker wins with high probability for large trees with bounded degree

Winning strategy exists within nearly linear moves

Results hold for a wide range of bias parameters q

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 $q \leq n^{\alpha}$. Moreover, we prove that Maker can win this game within $n + o(n)$ moves which is clearly asymptotically optimal.