Hitting time results for Maker-Breaker games

TL;DR

This paper determines the exact moments in a random graph process when Maker can guarantee winning strategies for connectivity, perfect matching, and Hamiltonicity, confirming previous conjectures.

Contribution

It provides optimal hitting time results for Maker-Breaker games on random graphs, settling existing conjectures and identifying precise thresholds for Maker's winning strategies.

Findings

Maker wins the $k$-vertex connectivity game at minimum degree $2k$

Maker wins the perfect matching game at minimum degree 2

Maker wins the Hamiltonicity game at minimum degree 4

Abstract

We study Maker-Breaker games played on the edge set of a random graph. Specifically, we consider the random graph process and analyze the first time in a typical random graph process that Maker starts having a winning strategy for his final graph to admit some property $\mP$. We focus on three natural properties for Maker's graph, namely being $k$-vertex-connected, admitting a perfect matching, and being Hamiltonian. We prove the following optimal hitting time results: with high probability Maker wins the $k$-vertex connectivity game exactly at the time the random graph process first reaches minimum degree $2k$; with high probability Maker wins the perfect matching game exactly at the time the random graph process first reaches minimum degree $2$; with high probability Maker wins the Hamiltonicity game exactly at the time the random graph process first reaches minimum degree $4$. The latter two statements settle conjectures of Stojakovi\'{c} and Szab\'{o}.