Powers of Hamilton Cycles in Pseudorandom Graphs

  • Peter Allen
  • Julia Böttcher
  • Hiệp Hàn
  • Yoshiharu Kohayakawa
  • Yury Person
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8392)


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, Y ⊂ V(G) with |X| ≥ εp k n and |Y| ≥ εp 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 [Triangle factors in sparse pseudo-random graphs, Combinatorica 24 (2004), no. 3, 403–426]. We also obtain results for higher powers of Hamilton cycles and establish corresponding counting versions. Our proofs are constructive, and yield deterministic polynomial time algorithms.


Random Graph Minimum Degree Hamilton Cycle Main Lemma Reservoir Property 
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Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  • Peter Allen
    • 1
  • Julia Böttcher
    • 1
  • Hiệp Hàn
    • 2
  • Yoshiharu Kohayakawa
    • 2
  • Yury Person
    • 3
  1. 1.Department of MathematicsLondon School of EconomicsLondonU.K.
  2. 2.Instituto de Matemática e EstatísticaUniversidade de São PauloSão PauloBrazil
  3. 3.Institute of MathematicsGoethe-UniversitätFrankfurtGermany

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