Advertisement

Combinatorica

, Volume 3, Issue 2, pp 181–192 | Cite as

Supersaturated graphs and hypergraphs

  • Paul Erdős
  • Miklós Simonovits
Article

Abstract

We shall consider graphs (hypergraphs) without loops and multiple edges. Let ℒ be a family of so called prohibited graphs and ex (n, ℒ) denote the maximum number of edges (hyperedges) a graph (hypergraph) onn vertices can have without containing subgraphs from ℒ. A graph (hyper-graph) will be called supersaturated if it has more edges than ex (n, ℒ). IfG hasn vertices and ex (n, ℒ)+k edges (hyperedges), then it always contains prohibited subgraphs. The basic question investigated here is: At least how many copies ofL ε ℒ must occur in a graphG n onn vertices with ex (n, ℒ)+k edges (hyperedges)?

AMS subject classification (1980)

05 C 35 05 C 65 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    B. Bollobás,Extremal graph theory, London Math. Soc. Monographs, No.11 Academic Press, 1978.Google Scholar
  2. [2]
    B. Bollobás. Relations between sets of complete subgraphs,Proc. Fifth British Comb. Conference, Aberdeen, 1975, 79–84.Google Scholar
  3. [3]
    B. Bollobás, On complete subgraphs of different orders,Math. Proc. Cambridge Phil. Soc. 79 (1976), 19–24.zbMATHCrossRefGoogle Scholar
  4. [4]
    B. Bollobás, P. Erdős andM. Simonovits. On the structure of edge graphs II,J. London Math. Soc. 12 (2) (1976) 219–224.zbMATHCrossRefGoogle Scholar
  5. [5]
    P. Erdős, On a theorem of Rademacher—Turán,Illinois J. Math. 6 (1962) 122–127.MathSciNetGoogle Scholar
  6. [6]
    P. Erdős, On the number of complete subgraphs contained in certain graphs,Publ. Math. Inst. Hungar. Acad. Sci. 7 (1962) 459–464.Google Scholar
  7. [7]
    P. Erdős, On extremal problems of graphs and generalized graphs,Israel J. Math. 2 (1965) 183–190.Google Scholar
  8. [8]
    P. Erdős, Some recent results on extremal problems in graph theory,Theory of Graphs, Intern. Symp. Rome, 1966, 118–123.Google Scholar
  9. [9]
    P. Erdős. On some new inequalities concerning extremal properties of graphs,Theory of Graphs, Proc. Coll. Tihany, 1966, 77–81.Google Scholar
  10. [10]
    P. Erdős, On the number of complete subgraphs and circuits contained in graphs,Casopis Pest. Mat. 94 (1969) 290–296.MathSciNetGoogle Scholar
  11. [11]
    P. Erdős, On some extremal problems onr-graphs,Discrete Math. 1 (1971) 1–6.CrossRefMathSciNetGoogle Scholar
  12. [12]
    P. Erdős andM. Simonovits, Compactness results in extremal graph theory,Combinatorica 2 (3) (1982), 275–288.CrossRefMathSciNetGoogle Scholar
  13. [13]
    P. Erdős andM. Simonovits, A limit theorem in graph theory,Studia Sci. Math. Hungar. 1 (1966) 51–57.MathSciNetGoogle Scholar
  14. [14]
    P. Erdős andM. Simonovits, Some extremal problems in graph theory,Coll. Math. Soc. János Bolyai 4 (1969) 377–390.Google Scholar
  15. [15]
    P. Erdős andA. H. Stone, On the structure of linear graphs, Bull. Amer. Math. Soc.52 (1946) 1089–1091.Google Scholar
  16. [16]
    Gy. Katona, T. Nemetz andM. Simonovits, A new proof on a theorem of Turán and some remarks on a generalization of it, (in Hungarian)Matematikai Lapok 15 (1964) 228–238.zbMATHMathSciNetGoogle Scholar
  17. [17]
    T. Kővári, V. T. Sós andP. Turán, On a problem of Zarankievicz,Coll. Math. 3 (1954) 50–57.Google Scholar
  18. [18]
    L. Lovász andM. Simonovits, On the number of complete subgraphs of a graph,Proc. Fifth British Combinatorial Coll., Aberdeen, (1975) 431–442.Google Scholar
  19. [19]
    L. Lovász andM. Simonovits, On the number of complete subgraphs of a graph, II.,Studies in Pure Mathematics, (1983), 459–495.Google Scholar
  20. [20]
    M. Simonovits, A method for solving extremal problems in graph theory,in: Theory of Graphs, Proc. Coll. Tihany, Hungary, 1966, 279–319.Google Scholar
  21. [21]
    P. Turán, On an extremal problem in graph theory, Mat. Fiz. Lapok48 (1941) 436–452.zbMATHMathSciNetGoogle Scholar
  22. [22]
    G. R. Blakley andP. Roy,Proc. Amer. Math. Soc. 16 (1965) 1244–1245; see alsoH. P. Mulfiolland andC. A. B. Smith,American Mathematical Monthly 66 (1959) 673–683, andD. London,Duke Math. Journal 33 (1966) 511–522.zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Akadémiai Kiadó 1983

Authors and Affiliations

  • Paul Erdős
    • 1
  • Miklós Simonovits
    • 1
  1. 1.Mathematical Institute of the Hungarian Academy of SciencesBudapestHungary

Personalised recommendations