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arXiv · 1008.1865

Hitting time results for Maker-Breaker games

Sonny Ben-Shimon·Asaf Ferber·Dan Hefetz·Michael Krivelevich
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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 Stojaković and Szabó.

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Sonny Ben-Shimon, Asaf Ferber, Dan Hefetz, Michael Krivelevich. 2010-08-11. Hitting time results for Maker-Breaker games. https://doi.org/10.1002/rsa.20392

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