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Corrádi and Hajnal's Theorem for Sparse Random Graphs

Published online by Cambridge University Press:  02 February 2012

JÓZSEF BALOGH ,
CHOONGBUM LEE  and
WOJCIECH SAMOTIJ
JÓZSEF BALOGH
Affiliation:
Department of Mathematics, University of Illinois, 1409 W Green Street, Urbana, IL 61801, USA and Department of Mathematics, University of California, San Diego, CA 92093, USA (e-mail: jobal@math.uiuc.edu)
CHOONGBUM LEE
Affiliation:
Department of Mathematics, UCLA, Los Angeles, CA 90095, USA (e-mail: choongbum.lee@gmail.com)
WOJCIECH SAMOTIJ
Affiliation:
School of Mathematical Sciences, Tel Aviv University, Tel Aviv 69978, Israel and Trinity College, Cambridge CB2 1TQ, UK (e-mail: samotij@post.tau.ac.il)
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Abstract

In this paper we extend a classical theorem of Corrádi and Hajnal into the setting of sparse random graphs. We show that if p(n) ≫ (log n/n)1/2, then asymptotically almost surely every subgraph of G(n, p) with minimum degree at least (2/3 + o(1))np contains a triangle packing that covers all but at most O(p−2) vertices. Moreover, the assumption on p is optimal up to the (log n)1/2 factor and the presence of the set of O(p−2) uncovered vertices is indispensable. The main ingredient in the proof, which might be of independent interest, is an embedding theorem which says that if one imposes certain natural regularity conditions on all three pairs in a balanced 3-partite graph, then this graph contains a perfect triangle packing.

Type
Paper
Information
Combinatorics, Probability and Computing , Volume 21 , Issue 1-2: Honouring the Memory of Richard H. Schelp , March 2012 , pp. 23 - 55
Copyright
Copyright © Cambridge University Press 2012

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References

[1]Alon, N. and Spencer, J. (2008) The Probabilistic Method, third edition, Wiley.CrossRefGoogle Scholar
[2]Alon, N. and Yuster, R. (1992) Almost H-factors in dense graphs. Graphs Combin. 8 95–102.CrossRefGoogle Scholar
[3]Alon, N. and Yuster, R. (1993) Threshold functions for H-factors. Combin. Probab. Comput. 2 137–144.CrossRefGoogle Scholar
[4]Alon, N. and Yuster, R. (1996) H-factors in dense graphs. J. Combin. Theory Ser. B 66 269–282.CrossRefGoogle Scholar
[5]Balogh, J., Csaba, B. and Samotij, W. (2011) Local resilience of almost spanning trees in random graphs. Random Struct. Alg. 38 121–139.CrossRefGoogle Scholar
[6]Ben-Shimon, S., Krivelevich, M. and Sudakov, B. (2011) Local resilience and Hamiltonicity Maker–Breaker games in random regular graphs. Combin. Probab. Comput. 20 173–211.CrossRefGoogle Scholar
[7]Böttcher, J., Kohayakawa, Y. and Taraz, A. (2009) Problem session. In Combinatorics and Probability Workshop, Oberwolfach.Google Scholar
[8]Böttcher, J., Kohayakawa, Y. and Taraz, A. Almost spanning subgraphs of random graphs after adversarial edge removal. arXiv:1003.0890v1 [math.CO]Google Scholar
[9]Corrádi, K. and Hajnal, A. (1963) On the maximal number of independent circuits in a graph. Acta Math. Acad. Sci. Hungar. 14 423–439.CrossRefGoogle Scholar
[10]Dellamonica, D., Kohayakawa, Y., Marciniszyn, M. and Steger, A. (2008) On the resilience of long cycles in random graphs. Electron. J. Combin. 15 #32.CrossRefGoogle Scholar
[11]Dirac, G. A. (1952) Some theorems on abstract graphs. Proc. London Math. Soc. 2 69–81.CrossRefGoogle Scholar
[12]Erdős, P. and Rényi, A. (1960) On the evolution of random graphs. Magyar Tud. Akad. Mat. Kutató Int. Közl. 5 17–61.Google Scholar
[13]Frieze, A. and Krivelevich, M. (2008) On two Hamilton cycle problems in random graphs. Israel J. Math. 166 221–234.CrossRefGoogle Scholar
[14]Gerke, S. and Steger, A. (2005) The sparse regularity lemma and its applications. In Surveys in Combinatorics 2005, Vol. 327 of London Mathematical Society Lecture Notes, Cambridge University Press, pp. 227–258.CrossRefGoogle Scholar
[15]Hajnal, A. and Szemerédi, E. (1970) Proof of a conjecture of P. Erdős. In Combinatorial Theory and its Applications II: Proc. Colloq., Balatonfüred, 1969, North-Holland, pp. 601–623.Google Scholar
[16]Hall, P. (1935) On representation of subsets. J. London Math. Soc. 10 26–30.CrossRefGoogle Scholar
[17]Huang, H., Lee, C. and Sudakov, B. (2012) Bandwidth theorem for random graphs. J. Comb. Theory B 102 14–37.CrossRefGoogle Scholar
[18]Johansson, A., Kahn, J. and Vu, V. (2008) Factors in random graphs. Random Struct. Alg. 33 1–28.CrossRefGoogle Scholar
[19]Kim, J. H. (2003) Perfect matchings in random uniform hypergraphs. Random Struct. Alg. 23 111–132.CrossRefGoogle Scholar
[20]Kohayakawa, Y. (1997) Szemerédi's regularity lemma for sparse graphs. In Foundations of Computational Mathematics, Rio de Janeiro, 1997, Springer, pp. 216–230.Google Scholar
[21]Kohayakawa, Y., Łuczak, T. and Rödl, V. (1997) On K 4-free subgraphs of random graphs. Combinatorica 17 173–213.CrossRefGoogle Scholar
[22]Kohayakawa, Y. and Rödl, V. (2003) Regular pairs in sparse random graphs I. Random Struct. Alg. 22 359–434.CrossRefGoogle Scholar
[23]Kohayakawa, Y. and Rödl, V. (2003) Szemerédi's regularity lemma and quasi-randomness. In Recent Advances in Algorithms and Combinatorics, Vol. 11 of CMS Books in Mathematics, Springer, pp. 289–351.Google Scholar
[24]Komlós, J., Sárközy, G. and Szemerédi, E. (1997) Blow-up lemma. Combinatorica 17 109–123.CrossRefGoogle Scholar
[25]Komlós, J., Sárközy, G. and Szemerédi, E. (2001) Proof of the Alon–Yuster conjecture. Discrete Math. 235 255–269.CrossRefGoogle Scholar
[26]Krivelevich, M. (1997) Triangle factors in random graphs. Combin. Probab. Comput. 6 337–347.CrossRefGoogle Scholar
[27]Krivelevich, M., Lee, C. and Sudakov, B. (2010) Resilient pancyclicity of random and pseudorandom graphs. SIAM J. Discrete Math. 24 1–16.CrossRefGoogle Scholar
[28]Krivelevich, M., Sudakov, B. and Szabó, T. (2004) Triangle factors in sparse pseudo-random graphs. Combinatorica 24 403–426.CrossRefGoogle Scholar
[29]Kühn, D. and Osthus, D. (2009) Embedding large subgraphs into dense graphs. In Surveys in combinatorics 2009, Vol. 365 of London Mathematical Society Lecture Notes, Cambridge University Press, pp. 137–167.CrossRefGoogle Scholar
[30]Kühn, D. and Osthus, D. (2009) The minimum degree threshold for perfect graph packings. Combinatorica 29 65–107.CrossRefGoogle Scholar
[31]Lee, C. and Sudakov, B. Dirac's theorem for random graphs. arXiv:1108.2502v2 [math.CO]Google Scholar
[32]Pósa, L. (1976) Hamiltonian circuits in random graphs. Discrete Math. 14 359–364.CrossRefGoogle Scholar
[33]Ruciński, A. (1992) Matching and covering the vertices of a random graph by copies of a given graph. Discrete Math. 105 185–197.CrossRefGoogle Scholar
[34]Sudakov, B. and Vu, V. H. (2008) Local resilience of graphs. Random Struct. Alg. 33 409–433.CrossRefGoogle Scholar
[35]Szemerédi, E. (1978) Regular partitions of graphs. In Problèmes Combinatoires et Théorie des Graphes: Orsay 1976, Vol. 260 of Colloq. Internat. CNRS, CNRS, pp. 399–401.Google Scholar