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Hamiltonian Berge cycles in random hypergraphs

Published online by Cambridge University Press:  08 September 2020

Deepak Bal
Affiliation:
Department of Mathematics, Montclair State University, Montclair, NJ 07043, USA
Ross Berkowitz
Affiliation:
Department of Mathematics, Yale University, New Haven, CT 06511, USA
Pat Devlin
Affiliation:
Department of Mathematics, Yale University, New Haven, CT 06511, USA
Mathias Schacht
Affiliation:
Department of Mathematics, University of Hamburg, 20146 Hamburg, Germany
Corresponding
E-mail address:

Abstract

In this note we study the emergence of Hamiltonian Berge cycles in random r-uniform hypergraphs. For $r\geq 3$ we prove an optimal stopping time result that if edges are sequentially added to an initially empty r-graph, then as soon as the minimum degree is at least 2, the hypergraph with high probability has such a cycle. In particular, this determines the threshold probability for Berge Hamiltonicity of the Erdős–Rényi random r-graph, and we also show that the 2-out random r-graph with high probability has such a cycle. We obtain similar results for weak Berge cycles as well, thus resolving a conjecture of Poole.

Type
Paper
Copyright
© The Author(s), 2020. Published by Cambridge University Press

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Footnotes

An earlier arXiv draft of this paper did not have our stopping time results.

References

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