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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
Abstract
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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 log3 n/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
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- © The Author(s), 2020. Published by Cambridge University Press
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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