Cycles and Unicyclic Components in Random Graphs

  • Authors:

  • Svante Janson

  • Affiliations:

  • Department of Mathematics, Uppsala University, PO Box 480, S-751 06 Uppsala, Sweden (e-mail: svante.janson@math.uu.se)

  • Venue:
  • Combinatorics, Probability and Computing
  • Year:
  • 2003

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Abstract

The sizes of the cycles and unicyclic components in the random graph $G(n, n/2\pm s)$, where $n^{2/3}\ll s \ll n$, are studied using the language of point processes. This refines several earlier results by different authors. Asymptotic distributions of various random variables are given: these distributions include the gamma distributions with parameters 1/4, 1/2 and 3/4, as well as the Poisson–Dirichlet and GEM distributions with parameters 1/4 and 1/2.