Supersaturated graphs and hypergraphs

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)?

This is a preview of subscription content, access via your institution.

References

  1. [1]

    B. Bollobás,Extremal graph theory, London Math. Soc. Monographs, No.11 Academic Press, 1978.

  2. [2]

    B. Bollobás. Relations between sets of complete subgraphs,Proc. Fifth British Comb. Conference, Aberdeen, 1975, 79–84.

  3. [3]

    B. Bollobás, On complete subgraphs of different orders,Math. Proc. Cambridge Phil. Soc. 79 (1976), 19–24.

    MATH  Article  Google 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.

    MATH  Article  Google Scholar 

  5. [5]

    P. Erdős, On a theorem of Rademacher—Turán,Illinois J. Math. 6 (1962) 122–127.

    MathSciNet  Google 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.

  9. [9]

    P. Erdős. On some new inequalities concerning extremal properties of graphs,Theory of Graphs, Proc. Coll. Tihany, 1966, 77–81.

  10. [10]

    P. Erdős, On the number of complete subgraphs and circuits contained in graphs,Casopis Pest. Mat. 94 (1969) 290–296.

    MathSciNet  Google Scholar 

  11. [11]

    P. Erdős, On some extremal problems onr-graphs,Discrete Math. 1 (1971) 1–6.

    Article  MathSciNet  Google Scholar 

  12. [12]

    P. Erdős andM. Simonovits, Compactness results in extremal graph theory,Combinatorica 2 (3) (1982), 275–288.

    Article  MathSciNet  Google Scholar 

  13. [13]

    P. Erdős andM. Simonovits, A limit theorem in graph theory,Studia Sci. Math. Hungar. 1 (1966) 51–57.

    MathSciNet  Google 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.

    MATH  MathSciNet  Google 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.

  19. [19]

    L. Lovász andM. Simonovits, On the number of complete subgraphs of a graph, II.,Studies in Pure Mathematics, (1983), 459–495.

  20. [20]

    M. Simonovits, A method for solving extremal problems in graph theory,in: Theory of Graphs, Proc. Coll. Tihany, Hungary, 1966, 279–319.

  21. [21]

    P. Turán, On an extremal problem in graph theory, Mat. Fiz. Lapok48 (1941) 436–452.

    MATH  MathSciNet  Google 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.

    MATH  Article  MathSciNet  Google Scholar 

Download references

Author information

Affiliations

Authors

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Erdős, P., Simonovits, M. Supersaturated graphs and hypergraphs. Combinatorica 3, 181–192 (1983). https://doi.org/10.1007/BF02579292

Download citation

AMS subject classification (1980)

  • 05 C 35
  • 05 C 65