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Powers of Hamilton cycles in pseudorandom graphs

Allen, Peter ORCID: 0000-0001-6555-3501, Böttcher, Julia ORCID: 0000-0002-4104-3635, Hàn, Hiệp, Kohayakawa, Yoshiharu and Person, Yury (2017) Powers of Hamilton cycles in pseudorandom graphs. Combinatorica, 37 (4). pp. 573-616. ISSN 0209-9683

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Identification Number: 10.1007/s00493-015-3228-2


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 and 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 Szab´o [Triangle factors in sparse pseudo-random graphs, Combinatorica 24 (2004), no. 3, 403–426]. We also extend our result to higher powers of Hamilton cycles and establish corresponding counting versions.

Item Type: Article
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Additional Information: © 2016 Springer International Publishing AG
Divisions: Mathematics
Subjects: Q Science > QA Mathematics
Date Deposited: 15 Apr 2015 15:53
Last Modified: 29 May 2024 04:42

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