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Finding tight Hamilton cycles in random hypergraphs faster

Part of: Graph theory

Published online by Cambridge University Press:  23 September 2020

Peter Allen ,
Christoph Koch ,
Olaf Parczyk  and
Yury Person
Peter Allen
Affiliation:
Department of Mathematics, London School of Economics, Houghton Street, London WC2A 2AE, UK
Christoph Koch
Affiliation:
Department of Statistics, University of Oxford, St Giles 24–29, Oxford OX1 3LB, UK
Olaf Parczyk*
Affiliation:
Department of Mathematics, London School of Economics, Houghton Street, London WC2A 2AE, UK
Yury Person
Affiliation:
Institut für Mathematik, Technische Universität Ilmenau, 98684 Ilmenau, Germany
*
*Corresponding author. Email: o.parczyk@lse.ac.uk
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Abstract

In an r-uniform hypergraph on n vertices, a tight Hamilton cycle consists of n edges such that there exists a cyclic ordering of the vertices where the edges correspond to consecutive segments of r vertices. We provide a first deterministic polynomial-time algorithm, which finds a.a.s. tight Hamilton cycles in random r-uniform hypergraphs with edge probability at least C log3n/n.

Our result partially answers a question of Dudek and Frieze, who proved that tight Hamilton cycles exist already for p = ω(1/n) for r = 3 and p = (e + o(1))/n for $r \ge 4$ using a second moment argument. Moreover our algorithm is superior to previous results of Allen, Böttcher, Kohayakawa and Person, and Nenadov and Škorić, in various ways: the algorithm of Allen et al. is a randomized polynomial-time algorithm working for edge probabilities $p \ge {n^{ - 1 + \varepsilon}}$, while the algorithm of Nenadov and Škorić is a randomized quasipolynomial-time algorithm working for edge probabilities $p \ge C\mathop {\log }\nolimits^8 n/n$.

MSC classification

Primary: 05C65: Hypergraphs
Secondary: 05C80: Random graphs 05C85: Graph algorithms
Type
Paper
Information
Combinatorics, Probability and Computing , Volume 30 , Issue 2 , March 2021 , pp. 239 - 257
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Copyright
© The Author(s), 2020. Published by Cambridge University Press

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Footnotes

†

The second author was supported by Austrian Science Fund (FWF), P26826, and European Research Council (ERC), no. 639046.

‡

The third and fourth authors were supported by DFG grant PE 2299/1-1.

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