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Largest Components in Random Hypergraphs

  • OLIVER COOLEY (a1), MIHYUN KANG (a1) and YURY PERSON (a2)
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.

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Combinatorics, Probability and Computing
  • ISSN: 0963-5483
  • EISSN: 1469-2163
  • URL: /core/journals/combinatorics-probability-and-computing
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