On the trace of random walks on random graphs

Alan Frieze, Michael Krivelevich, Peleg Michaeli, Ron Peled

Research output: Contribution to journalArticlepeer-review

Abstract

We study graph-theoretic properties of the trace of a random walk on a random graph. We show that for any ε > 0 there exists C > 0 such that the trace of the simple random walk of length (1 + ε)n ln n on the random graph G ∼ G(n, p) for p> C ln n/n is, with high probability, Hamiltonian and Θ(ln n)-connected. In the special case p = 1 (that is, when G = Kn), we show a hitting time result according to which, with high probability, exactly one step after the last vertex has been visited, the trace becomes Hamiltonian, and one step after the last vertex has been visited for the Kth time, the trace becomes 2k-connected.

Original languageEnglish
Pages (from-to)847-877
Number of pages31
JournalProceedings of the London Mathematical Society
Volume116
Issue number4
DOIs
StatePublished - Apr 2018

Keywords

  • 05C45
  • 05C80
  • 05C81 (primary)
  • 60G50 (secondary)

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