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Largest Components in Random Hypergraphs

Published online by Cambridge University Press:  04 April 2018

OLIVER COOLEY ,
MIHYUN KANG  and
YURY PERSON
OLIVER COOLEY
Affiliation:
Institute of Discrete Mathematics, Graz University of Technology, Steyrergasse 30, 8010 Graz, Austria (e-mail: cooley@math.tugraz.at, kang@math.tugraz.at)
MIHYUN KANG
Affiliation:
Institute of Discrete Mathematics, Graz University of Technology, Steyrergasse 30, 8010 Graz, Austria (e-mail: cooley@math.tugraz.at, kang@math.tugraz.at)
YURY PERSON
Affiliation:
Goethe-Universität, Institute of Mathematics, Robert-Mayer-Strasse 10, 60325 Frankfurt, Germany (e-mail: person@math.uni-frankfurt.de)
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Abstract

In this paper we consider j-tuple-connected components in random k-uniform hypergraphs (the j-tuple-connectedness relation can be defined by letting two j-sets be connected if they lie in a common edge and considering the transitive closure; the case j = 1 corresponds to the common notion of vertex-connectedness). We show that the existence of a j-tuple-connected component containing Θ(nj) j-sets undergoes a phase transition and show that the threshold occurs at edge probability

$$\frac{(k-j)!}{\binom{k}{j}-1}n^{j-k}.$$
Our proof extends the recent short proof for the graph case by Krivelevich and Sudakov, which makes use of a depth-first search to reveal the edges of a random graph.

Our main original contribution is a bounded degree lemma, which controls the structure of the component grown in the search process.

Keywords

Primary 05C65Secondary 05C80
Type
Paper
Information
Combinatorics, Probability and Computing , Volume 27 , Issue 5 , September 2018 , pp. 741 - 762
Copyright
Copyright © Cambridge University Press 2018 

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Footnotes

†

The first and third authors were supported by short visit grants 5639 and 5472, respectively, from the European Science Foundation (ESF) within the ‘Random Geometry of Large Interacting Systems and Statistical Physics’ (RGLIS) programme.

‡

The second author is supported by Austrian Science Fund (FWF): P26826, W1230, Doctoral Programme ‘Discrete Mathematics’.

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