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Counting Hamilton cycles in Dirac hypergraphs

Part of: Graph theory

Published online by Cambridge University Press:  17 December 2020

Stefan Glock ,
Stephen Gould ,
Felix Joos ,
Daniela Kühn  and
Deryk Osthus
Stefan Glock
Affiliation:
ETH Institute for Theoretical Studies, Clausiusstrasse 47, 8092 Zürich, Switzerland
Stephen Gould
Affiliation:
School of Mathematics, University of Birmingham, Edgbaston, Birmingham B15 2TT, UK
Felix Joos*
Affiliation:
Institut für Informatik, Heidelberg University, Germany
Daniela Kühn
Affiliation:
School of Mathematics, University of Birmingham, Edgbaston, Birmingham B15 2TT, UK
Deryk Osthus
Affiliation:
School of Mathematics, University of Birmingham, Edgbaston, Birmingham B15 2TT, UK
*
*Corresponding author. Email: joos@informatik.uni-heidelberg.de
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Abstract

A tight Hamilton cycle in a k-uniform hypergraph (k-graph) G is a cyclic ordering of the vertices of G such that every set of k consecutive vertices in the ordering forms an edge. Rödl, Ruciński and Szemerédi proved that for $k\ge 3$ , every k-graph on n vertices with minimum codegree at least $n/2+o(n)$ contains a tight Hamilton cycle. We show that the number of tight Hamilton cycles in such k-graphs is ${\exp(n\ln n-\Theta(n))}$ . As a corollary, we obtain a similar estimate on the number of Hamilton ${\ell}$ -cycles in such k-graphs for all ${\ell\in\{0,\ldots,k-1\}}$ , which makes progress on a question of Ferber, Krivelevich and Sudakov.

MSC classification

Primary: 05C65: Hypergraphs 05C81: Random walks on graphs 05C45: Eulerian and Hamiltonian graphs
Type
Paper
Information
Combinatorics, Probability and Computing , Volume 30 , Issue 4 , July 2021 , pp. 631 - 653
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Copyright
© The Author(s), 2020. Published by Cambridge University Press

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Footnotes

†

This project has received partial funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement 786198, D. Kühn and D. Osthus). The research leading to these results was also partially supported by the EPSRC, grants EP/N019504/1 (S. Glock and D. Kühn) and EP/S00100X/1 (D. Osthus), and the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation), grants 339933727 and 428212407 (F. Joos), as well as the Royal Society and the Wolfson Foundation (D. Kühn).

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