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The Asymptotic Number of Connected d-Uniform Hypergraphs

Published online by Cambridge University Press:  13 February 2014


MICHAEL BEHRISCH
Affiliation:
Institute of Transportation Systems, German Aerospace Center, Rutherfordstrasse 2, 12489 Berlin, Germany (e-mail: michael.behrisch@dlr.de)
AMIN COJA-OGHLAN
Affiliation:
Goethe University, Mathematics Institute, 60054 Frankfurt am Main, Germany (e-mail: acoghlan@math.uni-frankfurt.de)
MIHYUN KANG
Affiliation:
TU Graz, Institut für Optimierung und Diskrete Mathematik (Math B), Steyrergasse 30, 8010 Graz, Austria (e-mail: kang@math.tugraz.at)

Abstract

For d ≥ 2, let Hd(n,p) denote a random d-uniform hypergraph with n vertices in which each of the $\binom{n}{d}$ possible edges is present with probability p=p(n) independently, and let Hd(n,m) denote a uniformly distributed d-uniform hypergraph with n vertices and m edges. Let either H=Hd(n,m) or H=Hd(n,p), where m/n and $\binom{n-1}{d-1}p$ need to be bounded away from (d−1)−1 and 0 respectively. We determine the asymptotic probability that H is connected. This yields the asymptotic number of connected d-uniform hypergraphs with given numbers of vertices and edges. We also derive a local limit theorem for the number of edges in Hd(n,p), conditioned on Hd(n,p) being connected.


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Paper
Copyright
Copyright © Cambridge University Press 2014 

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Footnotes

An extended abstract version of this work appeared in the proceedings of RANDOM 2007, Vol. 4627 of Lecture Notes in Computer Science, Springer, pp. 341–352.


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