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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)

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, 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.

Keywords

Random Graph Minimum Degree Hamilton Cycle Main Lemma Reservoir Property 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    Allen, P., Böttcher, J., Hàn, H., Kohayakawa, Y., Person, Y.: Blow-up lemmas for sparse graphs (in preparation)Google Scholar
  2. 2.
    Allen, P., Böttcher, J., Kohayakawa, Y., Person, Y.: Tight Hamilton cycles in random hypergraphs. Random Structures Algorithms (to appear), doi: 10.1002/rsa.20519Google Scholar
  3. 3.
    Alon, N.: Explicit Ramsey graphs and orthonormal labelings. Electronic Journal of Combinatorics 1, Research paper 12, 8pp (1994)Google Scholar
  4. 4.
    Alon, N., Capalbo, M.: Sparse universal graphs for bounded-degree graphs. Random Structures Algorithms 31(2), 123–133 (2007)CrossRefzbMATHMathSciNetGoogle Scholar
  5. 5.
    Alon, N., Capalbo, M., Kohayakawa, Y., Rödl, V., Ruciński, A., Szemerédi, E.: Universality and tolerance (extended abstract). In: Proc. 41 IEEE FOCS, pp. 14–21. IEEE (2000)Google Scholar
  6. 6.
    Alon, N., Spencer, J.H.: The probabilistic method, vol. 57. Wiley Interscience (2000)Google Scholar
  7. 7.
    Bollobás, B.: The evolution of sparse graphs. In: Graph theory and combinatorics (Cambridge, 1983), pp. 35–57. Academic Press, London (1984)Google Scholar
  8. 8.
    Böttcher, J., Kohayakawa, Y., Taraz, A., Würfl, A.: An extension of the blow-up lemma to arrangeable graphs, arXiv:1305.2059Google Scholar
  9. 9.
    Chung, F.R.K., Graham, R.L., Wilson, R.M.: Quasi-random graphs. Combinatorica 9(4), 345–362 (1989)CrossRefzbMATHMathSciNetGoogle Scholar
  10. 10.
    Chung, F., Graham, R.: Sparse quasi-random graphs. Combinatorica 22(2), 217–244 (2002)CrossRefzbMATHMathSciNetGoogle Scholar
  11. 11.
    Conlon, D.: Talk at RSA 2013 (2013)Google Scholar
  12. 12.
    Conlon, D., Fox, J., Zhao, Y.: Extremal results in sparse pseudorandom graphs. arXiv:1204.6645Google Scholar
  13. 13.
    Cooper, C., Frieze, A.M.: On the number of Hamilton cycles in a random graph. J. Graph Theory 13(6), 719–735 (1989)CrossRefzbMATHMathSciNetGoogle Scholar
  14. 14.
    Dellamonica Jr., D., Kohayakawa, Y., Rödl, V., Ruciński, A.: An improved upper bound on the density of universal random graphs. In: Fernández-Baca, D. (ed.) LATIN 2012. LNCS, vol. 7256, pp. 231–242. Springer, Heidelberg (2012)CrossRefGoogle Scholar
  15. 15.
    Janson, S.: The numbers of spanning trees, Hamilton cycles and perfect matchings in a random graph. Combin. Probab. Comput. 3(1), 97–126 (1994)CrossRefzbMATHMathSciNetGoogle Scholar
  16. 16.
    Johansson, A., Kahn, J., Vu, V.: Factors in random graphs. Random Structures Algorithms 33(1), 1–28 (2008)zbMATHMathSciNetGoogle Scholar
  17. 17.
    Komlós, J., Sárközy, G.N., Szemerédi, E.: Blow-up lemma. Combinatorica 17(1), 109–123 (1997)CrossRefzbMATHMathSciNetGoogle Scholar
  18. 18.
    Komlós, J., Szemerédi, E.: Limit distribution for the existence of Hamiltonian cycles in a random graph. Discrete Math. 43(1), 55–63 (1983)CrossRefzbMATHMathSciNetGoogle Scholar
  19. 19.
    Korshunov, A.D.: Solution of a problem of Erdős and Renyi on Hamiltonian cycles in nonoriented graphs. Sov. Math., Dokl. 17, 760–764 (1976)zbMATHGoogle Scholar
  20. 20.
    Korshunov, A.D.: Solution of a problem of P. Erdős and A. Renyi on Hamiltonian cycles in undirected graphs. Metody Diskretn. Anal. 31, 17–56 (1977)zbMATHGoogle Scholar
  21. 21.
    Krivelevich, M.: On the number of Hamilton cycles in pseudo-random graphs. Electron. J. Combin. 19(1), Paper 25, 14pp (2012)Google Scholar
  22. 22.
    Krivelevich, M., Sudakov, B.: Sparse pseudo-random graphs are Hamiltonian. J. Graph Theory 42(1), 17–33 (2003)CrossRefzbMATHMathSciNetGoogle Scholar
  23. 23.
    Krivelevich, M., Sudakov, B.: Pseudo-random graphs, More sets, graphs and numbers. Bolyai Soc. Math. Stud. 15, 199–262 (2006)CrossRefMathSciNetGoogle Scholar
  24. 24.
    Krivelevich, M., Sudakov, B., Szabó, T.: Triangle factors in sparse pseudo-random graphs. Combinatorica 24(3), 403–426 (2004)CrossRefzbMATHMathSciNetGoogle Scholar
  25. 25.
    Kühn, D., Osthus, D.: On Pósa’s conjecture for random graphs. SIAM J. Discrete Math. 26(3), 1440–1457 (2012)Google Scholar
  26. 26.
    Pósa, L.: Hamiltonian circuits in random graphs. Discrete Mathematics 14(4), 359–364 (1976)CrossRefzbMATHMathSciNetGoogle Scholar
  27. 27.
    Riordan, O.: Spanning subgraphs of random graphs. Combin. Probab. Comput. 9(2), 125–148 (2000)CrossRefzbMATHMathSciNetGoogle Scholar
  28. 28.
    Thomason, A.: Pseudo-random graphs. Random Graphs 85, 307–331 (1987)MathSciNetGoogle Scholar

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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