Allen, Peter and Böttcher, Julia and Hàn, Hiệp and Kohayakawa, Yoshiharu and Person, Yury
(2016)
Powers of Hamilton cycles in pseudorandom graphs.
Combinatorica, 37 (4).
pp. 573616.
ISSN 02099683
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 and Y ⊆ V (G) with X ≥ εpkn and Y  ≥ εpℓn we have e(X, Y ) = (1 ± ε)pXY . 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 pseudorandom graphs, Combinatorica 24 (2004), no. 3, 403–426]. We also extend our result to higher powers of Hamilton cycles and establish corresponding counting versions.
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