Pancyclic subgraphs of random graphs

Choongbum Lee*, Wojciech Samotij

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

Abstract

An n-vertex graph is called pancyclic if it contains a cycle of length t for all 3> t >n. In this article, we study pancyclicity of random graphs in the context of resilience, and prove that if p>n-1/2, then the random graph G(n, p) a.a.s. satisfies the following property: Every Hamiltonian subgraph of G(n, p) with more than edges is pancyclic. This result is best possible in two ways. First, the range of p is asymptotically tight; second, the proportion of edges cannot be reduced. Our theorem extends a classical theorem of Bondy, and is closely related to a recent work of Krivelevich et al. The proof uses a recent result of Schacht (also independently obtained by Conlon and Gowers).

Original languageEnglish
Pages (from-to)142-158
Number of pages17
JournalJournal of Graph Theory
Volume71
Issue number2
DOIs
StatePublished - Oct 2012
Externally publishedYes

Keywords

  • Hamiltonicity
  • pancyclicity
  • random graph
  • resilience

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