Hitting time results for Maker-Breaker games

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 P. 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

• Type: Journal
• Year: 2010
• Where: CORR
• Authors: Sonny Ben-Shimon, Asaf Ferber, Dan Hefetz, Michael Krivelevich
• Keywords: CORR 2010, Education, Minimum Degree, Random Graph Process, Typical Random Graph