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.Comment: 30 pages, 1 figur