Hostname: page-component-cd9895bd7-7cvxr Total loading time: 0 Render date: 2024-12-22T16:02:33.853Z Has data issue: false hasContentIssue false
  • English
  • Français

An Upper Bound on Zarankiewicz' Problem

Published online by Cambridge University Press:  12 September 2008

Zoltán Füredi
Zoltán Füredi
Affiliation:
Department of Mathematics, University of Illinois at Urbana-Champaign, Urbana, IL 61801-2975, USA and Mathematical Institute of the Hungarian Academy of Sciences, PO Box 127, Budapest 1364, Hungary
Get access Rights & Permissions [Opens in a new window]

Abstract

Let ex(n, K3,3) denote the maximum number of edges of a K3,3-free graph on n vertices. Improving earlier results of Kővári, T. Sós and Turán on Zarankiewicz' problem, we obtain that Brown's example for a maximal K3,3-free graph is asymptotically optimal. Hence .

Type
Research Article
Information
Combinatorics, Probability and Computing , Volume 5 , Issue 1 , March 1996 , pp. 29 - 33
Copyright
Copyright © Cambridge University Press 1996

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Bollobás, B. (1978) Extremal Graph Theory, Academic Press.Google Scholar
[2]Brown, W. G. (1966) On graphs that do not contain a Thomsen graph. Canad. Math. Bull. 9 (1966), 281–289.CrossRefGoogle Scholar
[3]Erdős, P. and Rényi, A., (1960) On the evolution of random graphs. Magyar Tud. Akad. Mat. Kut. Int. Közl. 5, 17–61. (Also see The Art of Counting, Selected Writings of P. Erdős, J. Spencer ed., pp. 574–617. MIT Press, 1973).Google Scholar
[4]Erdős, P., Rényi, A. and Sós, V. T. (1966) On a problem of graph theory. Studia Sci. Math. Hungar. 1, 215–235.Google Scholar
[5]Erdős, P. and Simonovits, M. (1966) A limit theorem in graph theory. Studia Sci. Math. Hungar. 1, 51–57.Google Scholar
[6]Erdős, P. and Stone, A. H. (1946) On the structure of linear graphs. Bull. Amer. Math. Soc. 52, 1087–1091.CrossRefGoogle Scholar
[7]Füredi, Z. (1996) New asymptotics for bipartite Turän numbers. J. Combin. Th., Ser. A to appear.CrossRefGoogle Scholar
[8]Füredi, Z. (1996) On the number of edges of quadrilateral-free graphs. J. Combin. Th., Ser. B to appear.CrossRefGoogle Scholar
[9]Hyltén-Cavallius, C. (1958) On a combinatorial problem. Colloq. Math. 6, 59–65.CrossRefGoogle Scholar
[10]Kővári, T., Sós, V. T. and Turán, P. (1954) On a problem of K. Zarankiewicz. Colloq. Math. 3, 50–57.CrossRefGoogle Scholar
[11]Mantel, W. (1907) Problem 28. Wiskundige Opgaven 10 (1907), 60–61.Google Scholar
[12]Mörs, M. (1981) A new result on the problem of Zarankiewicz, J. Combin. Th., Ser. A 31, 126–130.CrossRefGoogle Scholar
[13]Turán, P. (1941) On an extremal problem in graph theory, Mat. Fiz. Lapok 48, 436–452 (in Hungarian). (Also see On the theory of graphs. Colloq. Math. 3, 19–30).Google Scholar
[14]Zarankiewicz, K. (1951) Problem of P101, Colloq. Math. 2, 301.Google Scholar
[15]Znám, Š. (1963) On a combinatorial problem of K. Zarankiewicz, Colloq. Math. 11, 81–84. (Also see Two improvements of a result concerning a problem of K. Zarankiewicz. Colloq. Math. 13 (1965), 255–258).CrossRefGoogle Scholar