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 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 [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.
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Allen, P., Böttcher, J., Hàn, H., Kohayakawa, Y., Person, Y.: Blow-up lemmas for sparse graphs (in preparation)
Allen, P., Böttcher, J., Kohayakawa, Y., Person, Y.: Tight Hamilton cycles in random hypergraphs. Random Structures Algorithms (to appear), doi: 10.1002/rsa.20519
Alon, N.: Explicit Ramsey graphs and orthonormal labelings. Electronic Journal of Combinatorics 1, Research paper 12, 8pp (1994)
Alon, N., Capalbo, M.: Sparse universal graphs for bounded-degree graphs. Random Structures Algorithms 31(2), 123–133 (2007)
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)
Alon, N., Spencer, J.H.: The probabilistic method, vol. 57. Wiley Interscience (2000)
Bollobás, B.: The evolution of sparse graphs. In: Graph theory and combinatorics (Cambridge, 1983), pp. 35–57. Academic Press, London (1984)
Böttcher, J., Kohayakawa, Y., Taraz, A., Würfl, A.: An extension of the blow-up lemma to arrangeable graphs, arXiv:1305.2059
Chung, F.R.K., Graham, R.L., Wilson, R.M.: Quasi-random graphs. Combinatorica 9(4), 345–362 (1989)
Chung, F., Graham, R.: Sparse quasi-random graphs. Combinatorica 22(2), 217–244 (2002)
Conlon, D.: Talk at RSA 2013 (2013)
Conlon, D., Fox, J., Zhao, Y.: Extremal results in sparse pseudorandom graphs. arXiv:1204.6645
Cooper, C., Frieze, A.M.: On the number of Hamilton cycles in a random graph. J. Graph Theory 13(6), 719–735 (1989)
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)
Janson, S.: The numbers of spanning trees, Hamilton cycles and perfect matchings in a random graph. Combin. Probab. Comput. 3(1), 97–126 (1994)
Johansson, A., Kahn, J., Vu, V.: Factors in random graphs. Random Structures Algorithms 33(1), 1–28 (2008)
Komlós, J., Sárközy, G.N., Szemerédi, E.: Blow-up lemma. Combinatorica 17(1), 109–123 (1997)
Komlós, J., Szemerédi, E.: Limit distribution for the existence of Hamiltonian cycles in a random graph. Discrete Math. 43(1), 55–63 (1983)
Korshunov, A.D.: Solution of a problem of Erdős and Renyi on Hamiltonian cycles in nonoriented graphs. Sov. Math., Dokl. 17, 760–764 (1976)
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)
Krivelevich, M.: On the number of Hamilton cycles in pseudo-random graphs. Electron. J. Combin. 19(1), Paper 25, 14pp (2012)
Krivelevich, M., Sudakov, B.: Sparse pseudo-random graphs are Hamiltonian. J. Graph Theory 42(1), 17–33 (2003)
Krivelevich, M., Sudakov, B.: Pseudo-random graphs, More sets, graphs and numbers. Bolyai Soc. Math. Stud. 15, 199–262 (2006)
Krivelevich, M., Sudakov, B., Szabó, T.: Triangle factors in sparse pseudo-random graphs. Combinatorica 24(3), 403–426 (2004)
Kühn, D., Osthus, D.: On Pósa’s conjecture for random graphs. SIAM J. Discrete Math. 26(3), 1440–1457 (2012)
Pósa, L.: Hamiltonian circuits in random graphs. Discrete Mathematics 14(4), 359–364 (1976)
Riordan, O.: Spanning subgraphs of random graphs. Combin. Probab. Comput. 9(2), 125–148 (2000)
Thomason, A.: Pseudo-random graphs. Random Graphs 85, 307–331 (1987)
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Allen, P., Böttcher, J., Hàn, H., Kohayakawa, Y., Person, Y. (2014). Powers of Hamilton Cycles in Pseudorandom Graphs. In: Pardo, A., Viola, A. (eds) LATIN 2014: Theoretical Informatics. LATIN 2014. Lecture Notes in Computer Science, vol 8392. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-54423-1_31
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