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|≥εp k n and |Y|≥εp ℓ n 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≪d 5/2 n -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 .
We also extend our result to higher powers of Hamilton cycles and establish corresponding counting versions.
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The cooperation of the authors was supported by a joint CAPES/DAAD project (415/ppp-probral/po/D08/11629, Proj. no. 333/09).
The authors are grateful to NUMEC/USP, Núcleo de Modelagem Estocástica e Complexidade of the University of São Paulo, and Project MaCLinC/USP, for supporting this research.
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Allen, P., Böttcher, J., Hàn, H. et al. Powers of Hamilton cycles in pseudorandom graphs. Combinatorica 37, 573–616 (2017). https://doi.org/10.1007/s00493-015-3228-2
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