Allen, Peter, Böttcher, Julia, Hàn, Hiệp, Kohayakawa, Yoshiharu and Person, Yury
(2014)
*Powers of hamilton cycles in pseudorandom graphs.*
LATIN 2014: Theoretical Informatics, 8392 (30).
pp. 355-366.
ISSN 1611-3349

## Abstract

We study the appearance of powers of Hamilton cycles in pseudorandom graphs, using the following comparatively weak pseudorandomness notion. A graph G is (ε, p, k, ℓ)-pseudorandom if for all disjoint X, Y ⊆ V (G) with /X/ ≥ εpkn and |Y| ≥ εpℓn we have e(X, Y ) = (1 ± ε)p/X//Y/. We prove that for all β > 0 there is an ε > 0 such that an (ε, p, 1, 2)-pseudorandom graph on n vertices with minimum degree at least βpn contains the square of a Hamilton cycle. In particular, this implies that (n, d, λ)-graphs with λ ≫ d5/2n?3/2 contain the square of a Hamilton cycle, and thus a triangle factor if n is a multiple of 3. This improves on a result of Krivelevich, Sudakov and Szabó [Triangle factors in sparse pseudo-random graphs, Combinatorica 24 (2004), no. 3, 403- 426]. We also obtain results for higher powers of Hamilton cycles and establish corresponding counting versions. Our proofs are constructive, and yield deterministic polynomial time algorithms.

Item Type: | Article |
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Official URL: | http://www.springer.com/computer/theoretical+compu... |

Additional Information: | © 2014 Springer-Verlag Berlin Heidelberg |

Divisions: | Mathematics |

Subjects: | G Geography. Anthropology. Recreation > GA Mathematical geography. Cartography |

JEL classification: | C - Mathematical and Quantitative Methods > C6 - Mathematical Methods and Programming > C61 - Optimization Techniques; Programming Models; Dynamic Analysis |

Sets: | Departments > Mathematics |

Date Deposited: | 09 Jun 2014 16:09 |

Last Modified: | 30 Jan 2019 19:34 |

URI: | http://eprints.lse.ac.uk/id/eprint/56949 |

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