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 14396912
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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