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Abstract

Let H d (n,p) signify a random d-uniform hypergraph with n vertices in which each of the \({n}\choose{d}\) possible edges is present with probability p = p(n) independently, and let H d (n,m) denote a uniformly distributed d-uniform hypergraph with n vertices and m edges. We establish a local limit theorem for the number of vertices and edges in the largest component of H d (n,p) in the regime, thereby determining the joint distribution of these parameters precisely. As an application, we derive an asymptotic formula for the probability that H d (n,m) is connected, thus obtaining a formula for the asymptotic number of connected hypergraphs with a given number of vertices and edges. While most prior work on this subject relies on techniques from enumerative combinatorics, we present a new, purely probabilistic approach.

Keywords

Joint Distribution Connected Graph Random Graph Large Component Random Structure 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2007

Authors and Affiliations

  • Michael Behrisch
    • 1
  • Amin Coja-Oghlan
    • 2
  • Mihyun Kang
    • 1
  1. 1.Humboldt-Universität zu Berlin, Institut für Informatik, Unter den Linden 6, 10099 BerlinGermany
  2. 2.Carnegie Mellon University, Department of Mathematical Sciences, Pittsburgh, PA 15213USA

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