Hamiltonicity thresholds in Achlioptas processes

Michael Krivelevich, Eyal Lubetzky, Benny Sudakov

Research output: Contribution to journalArticlepeer-review


In this article, we analyze the appearance of a Hamilton cycle in the following random process. The process starts with an empty graph on n labeled vertices. At each round we are presented with K = K(n) edges, chosen uniformly at random from the missing ones, and are asked to add one of them to the current graph. The goal is to create a Hamilton cycle as soon as possible. We show that this problem has three regimes, depending on the value of K. For K = o(logn), the threshold for Hamiltonicity is n logn, i.e., typically we can construct a Hamilton cycle K times faster that in the usual random graph process. When K = ω(logn) we can essentially waste almost no edges, and create a Hamilton cycle in n + o(n) rounds with high probability. Finally, in the intermediate regime where K = Θ (logn), the threshold has order n and we obtain upper and lower bounds that differ by a multiplicative factor of 3.

Original languageEnglish (US)
Pages (from-to)1-24
Number of pages24
JournalRandom Structures and Algorithms
Issue number1
StatePublished - Aug 2010


  • Hamilton cycles
  • Random graph processes

ASJC Scopus subject areas

  • Software
  • General Mathematics
  • Computer Graphics and Computer-Aided Design
  • Applied Mathematics


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