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Finding Tight Hamilton Cycles in Random Hypergraphs Faster

  • Peter Allen
  • Christoph Koch
  • Olaf Parczyk
  • Yury Person
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10807)

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 \(C \log ^3n/n\).

Our result partially answers a question of Dudek and Frieze (Random Struct Algorithms 42:374–385, 2013) who proved that tight Hamilton cycles exists already for \(p=\omega (1/n)\) 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 et al. (Random Struct Algorithms 46:446–465, 2015) and Nenadov and Škorić (arXiv:1601.04034) in various ways: the algorithm of Allen et al. is a randomised polynomial time algorithm working for edge probabilities \(p\ge n^{-1+\varepsilon }\), while the algorithm of Nenadov and Škorić is a randomised quasipolynomial time algorithm working for edge probabilities \(p\ge C\log ^8n/n\).

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Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  • Peter Allen
    • 1
  • Christoph Koch
    • 2
  • Olaf Parczyk
    • 3
  • Yury Person
    • 3
  1. 1.Department of MathematicsLondon School of EconomicsLondonUK
  2. 2.Mathematics InstituteUniversity of WarwickCoventryUK
  3. 3.Institut für MathematikGoethe UniversitätFrankfurt am MainGermany

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