Finding tight Hamilton cycles in random hypergraphs faster
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 Hamilto...
Saved in:
| Published in: | Combinatorics, probability & computing Vol. 30; no. 2; pp. 239 - 257 |
|---|---|
| Main Authors: | , , , |
| Format: | Journal Article |
| Language: | English |
| Published: |
Cambridge, UK
Cambridge University Press
01.03.2021
|
| Subjects: | |
| ISSN: | 0963-5483, 1469-2163 |
| Online Access: | Get full text |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| Summary: | 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 $p \ge C\mathop {\log }\nolimits^8 n/n$. |
|---|---|
| Bibliography: | ObjectType-Article-1 SourceType-Scholarly Journals-1 ObjectType-Feature-2 content type line 14 |
| ISSN: | 0963-5483 1469-2163 |
| DOI: | 10.1017/S0963548320000450 |