Hitting time results for Maker-Breaker games

Sonny Ben-Shimon, Asaf Ferber, Dan Hefetz, Michael Krivelevich

Research output: Contribution to journalArticlepeer-review

Abstract

We study Maker-Breaker games played on the edge set of a random graph. Specifically, we analyze the moment a typical random graph process first becomes a Maker's win in a game in which Maker's goal is to build a graph which admits some monotone increasing property P. We focus on three natural target 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. We also prove generalizations of the latter two results; these generalizations partially strengthen some known results in the theory of random graphs.

Original languageEnglish
Pages (from-to)23-46
Number of pages24
JournalRandom Structures and Algorithms
Volume41
Issue number1
DOIs
StatePublished - Aug 2012

Keywords

  • Hitting Time
  • Maker-Breaker Games
  • Random Graphs

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