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Regularity Lemmas for Graphs

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Fete of Combinatorics and Computer Science

Part of the book series: Bolyai Society Mathematical Studies ((BSMS,volume 20))

Abstract

Szemerédi’s regularity lemma proved to be a fundamental result in modern graph theory. It had a number of important applications and is a widely used tool in extremal combinatorics. For some further applications variants of the regularity lemma were considered. Here we discuss several of those variants and their relation to each other.

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References

  1. N. Alon, A. Coja-Oghlan, H. Hàn, M. Kang, V. Rödl and M. Schacht, Quasirandomness and algorithmic regularity for graphs with general degree distributions, SIAM J. Comput, to appear.

    Google Scholar 

  2. N. Alon, E. Fischer, M. Krivelevich and M. Szegedy, Efficient testing of large graphs, Combinatorica, 20 (2000), no. 4, 451–476.

    Article  MATH  MathSciNet  Google Scholar 

  3. N. Alon and A. Shapira, A characterization of the (natural) graph properties testable with one-sided error, SIAM J. Comput, 37 (2008), no. 6, 1703–1727.

    Article  MATH  MathSciNet  Google Scholar 

  4. N. Alon and A. Shapira, Every monotone graph property is testable, SIAM J. Comput, 38 (2008), no. 2, 505–522.

    Article  MathSciNet  Google Scholar 

  5. T. Austin and T. Tao, On the testability and repair of hereditary hypergraph properties, submitted.

    Google Scholar 

  6. W. G. Brown, P. Erdős and V. T. Sós, Some extremal problems on r-graphs, New directions in the theory of graphs (Proc. Third Ann Arbor Conf., Univ. Michigan, Ann Arbor, Mich, 1971), Academic Press, New York, 1973, pp. 53–63.

    Google Scholar 

  7. V. Chvátal, V. Rödl, E. Szemerédi and W. T. Trotter, Jr., The Ramsey number of a graph with bounded maximum degree, J. Combin. Theory Ser. B, 34 (1983), no. 3, 239–243.

    Article  MATH  MathSciNet  Google Scholar 

  8. R. A. Duke, H. Lefmann and V. Rödl, A fast approximation algorithm for computing the frequencies of subgraphs in a given graph, SIAM J. Comput, 24 (1995), no. 3, 598–620.

    Article  MATH  MathSciNet  Google Scholar 

  9. P. Erdös and P. Turán, On some sequences of integers, J. London. Math. Soc, 11 (1936), 261–264.

    Article  MATH  Google Scholar 

  10. P. Erdös, P. Frankl and V. Rödl, The asymptotic number of graphs not containing a fixed subgraph and a problem for hypergraphs having no exponent, Graphs Combin., 2 (1986), no. 2, 113–121.

    Article  MATH  MathSciNet  Google Scholar 

  11. P. Frankl and V. Rödl, Extremal problems on set systems, Random Structures Algorithms, 20 (2002), no. 2, 131–164.

    Article  MATH  MathSciNet  Google Scholar 

  12. A. Frieze and R. Kannan, Quick approximation to matrices and applications, Combinatorica, 19 (1999), no. 2, 175–220.

    Article  MATH  MathSciNet  Google Scholar 

  13. H. Furstenberg and Y. Katznelson, An ergodic Szemerédi theorem for commuting transformations, J. Analyse Math., 34 (1978), 275–291 (1979).

    Article  MATH  MathSciNet  Google Scholar 

  14. H. Furstenberg and Y. Katznelson, An ergodic Szemerédi theorem for IP-systems and combinatorial theory, J. Analyse Math., 45 (1985), 117–168.

    Article  MATH  MathSciNet  Google Scholar 

  15. S. Gerke and A. Steger, The sparse regularity lemma and its applications, Surveys in combinatorics 2005, London Math. Soc. Lecture Note Ser., vol. 327, Cambridge Univ. Press, Cambridge, 2005, pp. 227–258.

    Google Scholar 

  16. O. Goldreich, S. Goldwasser and D. Ron, Property testing and its connection to learning and approximation, J. ACM, 45 (1998), no. 4, 653–750.

    Article  MATH  MathSciNet  Google Scholar 

  17. W. T. Gowers, Lower bounds of tower type for Szemerédi’s uniformity lemma, Geom. Funct. Anal, 7 (1997), no. 2, 322–337.

    Article  MATH  MathSciNet  Google Scholar 

  18. W. T. Gowers, Hypergraph regularity and the multidimensional Szemerédi theorem, Ann. of Math. (2), 166 (2007), no. 3, 897–946.

    Article  MathSciNet  Google Scholar 

  19. P. E. Haxell, Y. Kohayakawa and T. Luczak, The induced size-Ramsey number of cycles, Combin. Probab. Comput, 4 (1995), no. 3, 217–239.

    Article  MATH  MathSciNet  Google Scholar 

  20. P. E. Haxell, Y. Kohayakawa and T. Luczak, Turán’s extremal problem in random graphs: forbidding even cycles, J. Combin. Theory Ser. B, 64 (1995), no. 2, 273–287.

    Article  MATH  MathSciNet  Google Scholar 

  21. P. E. Haxell, Y. Kohayakawa and T. Luczak, Turán’s extremal problem in random graphs: forbidding odd cycles, Combinatorica, 16 (1996), no. 1, 107–122.

    Article  MATH  MathSciNet  Google Scholar 

  22. Y. Kohayakawa, Szemerédi’s regularity lemma for sparse graphs, Foundations of computational mathematics (Rio de Janeiro, 1997), pp. 216–230.

    Google Scholar 

  23. Y. Kohayakawa and V. Rödl, Szemerédi’s regularity lemma and quasi-randomness, Recent advances in algorithms and combinatorics, CMS Books Math./Ouvrages Math. SMC, vol. 11, Springer, New York, 2003, pp. 289–351.

    Google Scholar 

  24. J. Komlös, G. N. Sárközy and E. Szemerédi, Blow-up lemma, Combinatorica, 17 (1997), no. 1, 109–123.

    Article  MathSciNet  Google Scholar 

  25. J. Komlós, A. Shokoufandeh, M. Simonovits and E. Szemerédi, The regularity lemma and its applications in graph theory, Theoretical aspects of computer science (Tehran, 2000), Lecture Notes in Comput. Sci., vol. 2292, Springer, Berlin, 2002, pp. 84–112.

    Google Scholar 

  26. J. Komlös and M. Simonovits, Szemerédi’s regularity lemma and its applications in graph theory, Combinatorics, Paul Erdős is eighty, Vol. 2 (Keszthely, 1993), Bolyai Soc. Math. Stud., vol. 2, János Bolyai Math. Soc, Budapest, 1996, pp. 295–352.

    Google Scholar 

  27. L. Lovász and B. Szegedy, Testing properties of graphs and functions, Israel J. Math., to appear.

    Google Scholar 

  28. L. Lovász and B. Szegedy, Graph limits and testing hereditary graph properties, Tech. Report MSR-TR-2005-110, Microsoft Research, 2005.

    Google Scholar 

  29. L. Lovász and B. Szegedy, Limits of dense graph sequences, J. Combin. Theory Ser. B, 96 (2006), no. 6, 933–957.

    Article  MATH  MathSciNet  Google Scholar 

  30. L. Lovasz and B. Szegedy, Szemerédi’s lemma for the analyst, Geom. Funct. Anal, 17 (2007), no. 1, 252–270.

    Article  MATH  MathSciNet  Google Scholar 

  31. T. Luczak, Randomness and regularity, International Congress of Mathematicians. Vol. III, Eur. Math. Soc, Zürich, 2006, pp. 899–909.

    Google Scholar 

  32. B. Nagle and V. Rödl, Regularity properties for triple systems, Random Structures Algorithms, 23 (2003), no. 3, 264–332.

    Article  MATH  MathSciNet  Google Scholar 

  33. B. Nagle, V. Rödl and M. Schacht, The counting lemma for regular k-uniform hypergraphs, Random Structures Algorithms, 28 (2006), no. 2, 113–179.

    Article  MATH  MathSciNet  Google Scholar 

  34. V. Rödl, unpublished.

    Google Scholar 

  35. V. Rödl, Some developments in Ramsey theory, Proceedings of the International Congress of Mathematicians, Vol. I, II (Kyoto, 1990) (Tokyo), Math. Soc. Japan, 1991, pp. 1455–1466.

    Google Scholar 

  36. V. Rödl and M. Schacht, Generalizations of the removal lemma, Combinatorica, 29 (2009), no. 4, 467–501.

    Article  MATH  MathSciNet  Google Scholar 

  37. V. Rödl and M. Schacht, Regular partitions of hypergraphs: regularity lemmas, Combin. Probab. Comput., 16 (2007), no. 6, 833–885.

    MATH  MathSciNet  Google Scholar 

  38. V. Rödl, M. Schacht, E. Tengan and N. Tokushige, Density theorems and extremal hypergraph problems, Israel J. Math., 152 (2006), 371–380.

    Article  MATH  MathSciNet  Google Scholar 

  39. V. Rödl and J. Skokan, Regularity lemma for k-uniform hypergraphs, Random Structures Algorithms, 25 (2004), no. 1, 1–42.

    Article  MATH  MathSciNet  Google Scholar 

  40. V. Rödl and J. Skokan, Counting subgraphs in quasi-random 4-uniform hypergraphs, Random Structures Algorithms, 26 (2005), no. 1-2, 160–203.

    Article  MATH  MathSciNet  Google Scholar 

  41. V. Rödl and J. Skokan, Applications of the regularity lemma for uniform hypergraphs, Random Structures Algorithms, 28 (2006), no. 2, 180–194.

    Article  MATH  MathSciNet  Google Scholar 

  42. K. F. Roth, On certain sets of integers, J. London Math. Soc, 28 (1953), 104–109.

    Article  MATH  MathSciNet  Google Scholar 

  43. I. Z. Ruzsa and E. Szemerédi, Triple systems with no six points carrying three triangles, Combinatorics (Proc. Fifth Hungarian Colloq., Keszthely, 1976), Vol. II, Colloq. Math. Soc. János Bolyai, vol. 18, North-Holland, Amsterdam, 1978, pp. 939–945.

    Google Scholar 

  44. J. Solymosi, A note on a question of Erdős and Graham, Combin. Probab. Comput, 13 (2004), no. 2, 263–267.

    Article  MATH  MathSciNet  Google Scholar 

  45. V. T. Sös, P. Erdős and W. G. Brown, On the existence of triangulated spheres in 3-graphs, and related problems, Period. Math. Hungar., 3 (1973), no. 3-4, 221–228.

    Article  MathSciNet  Google Scholar 

  46. E. Szemerédi, On graphs containing no complete subgraph with 4 vertices, Mat. Lapok, 23 (1972), 113–116 (1973).

    MathSciNet  Google Scholar 

  47. E. Szemerédi, On sets of integers containing no k elements in arithmetic progression, Acta Arith., 27 (1975), 199–245, Collection of articles in memory of Jurii Vladimirovič Linnik.

    MATH  MathSciNet  Google Scholar 

  48. E. Szemerédi, Regular partitions of graphs, Problèmes combinatoires et théorie des graphes (Colloq. Internat. CNRS, Univ. Orsay, Orsay, 1976), Colloq. Internat. CNRS, vol. 260, CNRS, Paris, 1978, pp. 399–401.

    Google Scholar 

  49. T. Tao, Szemerédi’s regularity lemma revisited, Contrib. Discrete Math., 1 (2006), no. 1, 8–28 (electronic).

    MATH  MathSciNet  Google Scholar 

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Author information

Authors and Affiliations

  1. Department of Mathematics and Computer Science, Emory University, Atlanta, GA 30322, USA

    Vojtěch Rödl

  2. Institut für Informatik, Humboldt- Universität zu Berlin, Unter den Linden 6, D-10099 Berlin, Germany

    Mathias Schacht

Authors
  1. Vojtěch Rödl
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  2. Mathias Schacht
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Editor information

Editors and Affiliations

  1. Alfréd Rényi Institute of Mathematics, Hungarian Academy of Sciences, Reáltanoda u. 13-15, Budapest, 1053, Hungary

    Gyula O. H. Katona

  2. Centre for Mathematics and Computer Science, Science Park 123, 1098 XG, Amsterdam, The Netherlands

    Alexander Schrijver

  3. Department of Computer Science, Eötvös University, Pázmány Péter sétány 1/C, Budapest, 1117, Hungary

    Tamás Szőnyi

  4. Alfréd Rényi Institute of Mathematics, Hungarian Academy of Sciences, Reáltanoda u. 13-15, Budapest, 1053, Hungary

    Gábor Sági

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Rödl, V., Schacht, M. (2010). Regularity Lemmas for Graphs. In: Katona, G.O.H., Schrijver, A., Szőnyi, T., Sági, G. (eds) Fete of Combinatorics and Computer Science. Bolyai Society Mathematical Studies, vol 20. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-13580-4_11

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