Fast strategies in Maker-Breaker games played on random boards

TL;DR

This paper studies Maker-Breaker games on sparse random graphs, showing Maker can win key games quickly when the graph's edge probability exceeds a polylogarithmic threshold.

Contribution

It establishes near-optimal winning times for Maker in Hamiltonicity, perfect matching, and k-connectivity games on random graphs with certain edge probabilities.

Findings

Maker wins these games in near-linear time on sparse random graphs.

Winning strategies are effective when the edge probability exceeds polylogarithmic thresholds.

Results apply to Hamiltonicity, perfect matching, and k-connectivity games.

Abstract

In this paper we analyze classical Maker-Breaker games played on the edge set of a sparse random board $G\sim \gnp$. We consider the Hamiltonicity game, the perfect matching game and the $k$-connectivity game. We prove that for $p(n)\geq \text{polylog}(n)/n$, the board $G\sim \gnp$ is typically such that Maker can win these games asymptotically as fast as possible, i.e. within $n+o(n)$, $n/2+o(n)$ and $kn/2+o(n)$ moves respectively.