A sharp threshold for the hamilton cycle Maker-Breaker game

Dan Hefetz*, Michael Krivelevich, Miloš Stojakovič, Tibor Szabó

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review


We study the Hamilton cycle Maker-Breaker game, played on the edges of the random graph G(n,p). We prove a conjecture from (Stojaković and Szabó, Random Struct and Algorithms 26 (2005), 204-223.), asserting that the property that Maker is able to win this game, has a sharp threshold at log n/n. Our theorem can be considered a game-theoretic strengthening of classical results from the theory of random graphs: not only does G(n,p) almost surely admit a Hamilton cycle for p = (1 + ε) log n/n, but Maker is able to build one while playing against an adversary.

Original languageEnglish
Pages (from-to)112-122
Number of pages11
JournalRandom Structures and Algorithms
Issue number1
StatePublished - Jan 2009


  • Combinatorial games
  • Hamilton cycle
  • Random graph


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