Finding tight Hamilton cycles in random hypergraphs faster. (23rd March 2021)

• Record Type: Journal Article
• Authors: Allen, Peter; Koch, Christoph; Parczyk, Olaf; Person, Yury
• Is Part Of: Combinatorics, probability and computing. Volume 30:Number 2(2021)
• Page Start: 239
• Page End: 257
• Publication Date: 2021-03-23
• Subjects: 05C65, -- 05C80, -- 05C85; Combinatorial analysis -- Periodicals; Probabilities -- Periodicals; Computer science -- Mathematics -- Periodicals; 511.6
• DOI: 10.1017/S0963548320000450
• Languages: English
• ISSNs: 0963-5483

Abstract

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 3 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 $p \ge C\mathop {\log }\nolimits^8 n/n$ .