Abstract
In Combinatorica 17(2), 1997, Kohayakawa, Łuczak and Rödl state a conjecture which has several implications for random graphs. If the conjecture is true, then, for example, an application of a version of Szemerédi’s regularity lemma for sparse graphs yields an estimation of the maximal number of edges in an H-free subgraph of a random graph G n, p . In fact, the conjecture may be seen as a probabilistic embedding lemma for partitions guaranteed by a version of Szemerédi’s regularity lemma for sparse graphs. In this paper we verify the conjecture for H = K 4, thereby providing a conceptually simple proof for the main result in the paper cited above.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
M. Behrisch: Random graphs without a short cycle, Master’s thesis, Humboldt-Universität zu Berlin, 2002 (in German).
Z. Füredi: Random Ramsey graphs for the four-cycle, Discrete Mathematics 126 (1994), 407–410.
S. Gerke, Y. Kohayakawa, V. R ödl and A. Steger: Small subsets inherit sparse ɛ-regularity, Journal of Combinatorial Theory B, accepted. http://dx.doi.org/10.1016/j.jctb.2006.03.004
S. Gerke, M. Marciniszyn and A. Steger: A probabilistic counting lemma for complete graphs, in: 2005 European Conference on Combinatorics, Graph Theory and Applications (EuroComb’ 05), S. Felsner, ed., pp. 309–316, DMTCS Proceedings, 2005 (extended abstract). To appear in Random Structures & Algorithms.
S. Gerke and A. Steger: The sparse regularity lemma and its applications, in: Surveys in Combinatorics 2005, LMS, B. S. Webb, ed., LNS 327, pp. 227–258, CUP, 2005.
S. Gerke, T. Schickinger and A. Steger: K 5-free subgraphs of random graphs, Random Structures Algorithms 24(2) (2004), 194–232.
P. E. Haxell, Y. Kohayakawa and T. Łuczak: Turán’s extremal problem in random graphs: forbidding even cycles, Journal of Combinatorial Theory B 64 (1995), 273–287.
P. E. Haxell, Y. Kohayakawa and T. Łuczak: Turán’s extremal problem in random graphs: forbidding odd cycles; Combinatorica 16(1) (1996), 107–122.
Y. Kohayakawa and B. Kreuter: Threshold functions for asymmetric Ramsey properties involving cycles, Random Structures & Algorithms 11 (1997), 245–276.
Y. Kohayakawa, B. Kreuter and A. Steger: An extremal problem for random graphs and the number of graphs with large even girth, Combinatorica 18(1) (1998), 101–120.
Y. Kohayakawa, T. Łuczak and V. Rödl: Arithmetic progressions of length three in subsets of a random set, Acta Arithmetica LXXV.2 (1996), 133–163.
Y. Kohayakawa, T. Łuczak and V. Rödl: On K 4-free subgraphs of random graphs, Combinatorica 17(2) (1997), 173–213.
Y. Kohayakawa and V. Rödl: Regular pairs in sparse random graphs I, Random Structures & Algorithms 22(4) (2003), 359–434.
B. Kreuter: Probabilistic versions of Ramsey’s and Turán’s theorems. PhD thesis, Humboldt-Universität zu Berlin, 1997.
Y. Kohayakawa, V. Rödl and M. Schacht: The Turán theorem for random graphs, Combin. Probab. Comput. 13(1) (2004), 61–91.
D. J. Kleitman and K. J. Winston: On the number of graphs without 4-cycles, Discrete Mathematics 41 (1982), 167–172.
T. Łuczak: On triangle-free random graphs, Random Structures & Algorithms 16(3) (2000), 260–276.
T. Szabó and V. H. Vu: Turán’s theorem in sparse random graphs, Random Structures & Algorithms 23(3) (2003), 225–234.
Author information
Authors and Affiliations
Corresponding author
Additional information
Supported by DFG-Grant Ste 464/3-1.
Rights and permissions
About this article
Cite this article
Gerke, S., Prömel, H.J., Schickinger, T. et al. K 4-free subgraphs of random graphs revisited. Combinatorica 27, 329–365 (2007). https://doi.org/10.1007/s00493-007-2010-5
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00493-007-2010-5