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 \(p\ge C\log ^8n/n\).
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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Notes
- 1.
Writing \(\mathcal {O}^*\) means we ignore polylogarithmic factors.
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Allen, P., Koch, C., Parczyk, O., Person, Y. (2018). Finding Tight Hamilton Cycles in Random Hypergraphs Faster. In: Bender, M., Farach-Colton, M., Mosteiro, M. (eds) LATIN 2018: Theoretical Informatics. LATIN 2018. Lecture Notes in Computer Science(), vol 10807. Springer, Cham. https://doi.org/10.1007/978-3-319-77404-6_3
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