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Combinatorica

, Volume 2, Issue 3, pp 275–288 | Cite as

Compactness results in extremal graph theory

  • P. Erdős
  • M. Simonovits
Article

Abstract

Let L be a given family of so called prohibited graphs. Let ex (n, L) denote the maximum number of edges a simple graph of ordern can have without containing subgraphs from L. A typical extremal graph problem is to determine ex (n, L), or at least, find good bounds on it. Results asserting that for a given L there exists a much smaller L*⫅L for which ex (n, L) ≈ ex (n, L*) will be calledcompactness results. The main purpose of this paper is to prove some compactness results for the case when L consists of cycles. One of our main tools will be finding lower bounds on the number of pathsP k+1 in a graph ofn vertices andE edges., witch is, in fact, a “super-saturated” version of a wellknown theorem of Erdős and Gallai.

AMS subject classification (1980)

05 C 35 
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References

  1. [1]
    C. Benson, Minimal regular graphs of girth eight and twelve,Canadian J. Math. 18 (1966) 1091–1094.zbMATHMathSciNetGoogle Scholar
  2. [2]
    J. A. Bondy andM. Simonovits, Cycles of even length in graphs,Journal of Combinatorial Theory,16 (1974) 97–105.zbMATHCrossRefMathSciNetGoogle Scholar
  3. [3]
    W. G. Brown, On graphs that do not contain a Thomsen graph,Canadian Math. Bull. 9 (1966) 281–285.zbMATHGoogle Scholar
  4. [4]
    W. G. Brown, P. Erdős andM. Simonovits, Algorithmic solution of extremal digraph problems,to be published.Google Scholar
  5. [5]
    D. Cvetković, M. Doob andH. Sachs,Spectra of graphs — Theory and Application — VEB Deuthscher Verlag der Wissenschaften, Berlin, 1980.Google Scholar
  6. [6]
    P. Erdős, On sequences of integers no one of which divides the product of two others, and some related problems,Mitt. Forschunginstitut Math. u. Mech. Tomsk 2 (1938) 74–82.Google Scholar
  7. [7]
    P. Erdős, Graph theory and probability,Can J. Math. 11 (1959) 34–38.Google Scholar
  8. [8]
    P. Erdős, A. Rényi andV. T. Sós, On a problem in graph theory,Studia Sci. Math. Hungar. 1 (1966), 215–235.MathSciNetGoogle Scholar
  9. [9]
    P. Erdős andM. Simonovits, A limit theorem in graph theoryStudia Sci. Math. Hungar. 1 (1966) 51–57.MathSciNetGoogle Scholar
  10. [10]
    P. Erdős andM. Simonovits, Supersaturated graphs,submitted to Combinatorica.Google Scholar
  11. [11]
    G. R. Blakley andP. Roy Hölder type inequality for symmetric matrices with nonnegative entries.Proc. AMS 16 (1965) 1244–1245.zbMATHCrossRefMathSciNetGoogle Scholar
  12. [12]
    R. J. Faudree andM. Simonovits, On a class of degenerate extremal problems,submitted to Combinatorica.Google Scholar
  13. [13]
    T. Kővári, V. T. Sós andP. Turán, On a problem of Zarankiewicz,Coll. Math. 3 (1954) (50–57).Google Scholar
  14. [14]
    I. Reiman, Über ein problem von Zarankiewicz,Acta Acad. Sci. Hungar. 9 (1958) 269–279.zbMATHCrossRefMathSciNetGoogle Scholar
  15. [15]
    R. Singleton, On minimal graphs of maximum even girth,J. Combin. Theory 1 (1966) 306–332.zbMATHMathSciNetGoogle Scholar
  16. [16]
    P. Turán, On an extremal problem in graph theory, (in Hungarian),Mat. Fiz. Lapok 48 (1941) 436–452.zbMATHMathSciNetGoogle Scholar
  17. [17]
    D. London, Inequalities in quadratic forms,Duke Math. J. 83 (1966) 511–522.CrossRefMathSciNetGoogle Scholar
  18. [18]
    H. P. Mulholland andC. A. B. Smith, An inequality arising in genetical theory,Amer. Math. Monthly 66 (1969), 673–683.CrossRefMathSciNetGoogle Scholar

Copyright information

© Akadémiai Kiadó 1982

Authors and Affiliations

  • P. Erdős
    • 1
  • M. Simonovits
    • 2
  1. 1.Mathematical Institute of the Hungarian Academy of SciencesBudapestHungary
  2. 2.Dept. Analysis I.Eötvös UniversityBudapestHungary

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