Skip to main content
SpringerLink
Log in
Book cover

The Mathematics of Paul Erdös II pp 70–78Cite as

Hereditary and Monotone Properties of Graphs

  • Chapter

Part of the book series: Algorithms and Combinatorics ((AC,volume 14))

Summary

Given a hereditary graph property Ρ let Ρ n be the set of those graphs in Ρ on the vertex set {1, …, n}. Define the constant c n by \(\left| {P^n } \right| = 2^{cn(_2^n )} .\) We show that the limit limn → ∞ c n always exists and equals 1 − 1/r, where r is a positive integer which can be described explicitly in terms of Ρ. This result, obtained independently by Alekseev, extends considerably one of Erdős, Frankl and Rödl concerning principal monotone properties and one of Prömel and Steger concerning principal hereditary properties.

AMS Subject Classification: Primary 05C35, Secondary 05C30.

This is a preview of subscription content, log in via an institution.

Chapter
USD   29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   74.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. V.E. Alekseev, Hereditary classes and coding of graphs, Probl. Cybern. 39 (1982) 151–164 (in Russian).

    Google Scholar 

  2. V.E. Alekseev, On the entropy values of hereditary classes of graphs, Discrete Math. Appi. 3 (1993) 191–199.

    Article  Google Scholar 

  3. B. Bollobás, Extremal Graph Theory, Academic Press (1978), xx+488pp.

    MATH  Google Scholar 

  4. B. Bollobás and P. Erdős, On the structure of edge graphs, Bull. London Math. Soc. 5 (1973) 317–321.

    Article  MathSciNet  MATH  Google Scholar 

  5. B. Bollobás, P. Erdős and M. Simonovits, On the structure of edge graphs II, J. London Math. Soc. 12(2) (1976) 219–224.

    Article  MATH  Google Scholar 

  6. B. Bollobás and A. Thomason, Projections of bodies and hereditary properties of hypergraphs (submitted).

    Google Scholar 

  7. B. Bollobás and A. Thomason, An extension of the Erdős-Stone theorem, (in preparation).

    Google Scholar 

  8. V. Chvátal and E. Szemerédi, On the Erdős-Stone theorem, J. London Math. Soc. 23 (1981) 207–214.

    Article  MathSciNet  MATH  Google Scholar 

  9. P. Erdős, P. Frankl and V. Rödl, The asymptotic enumeration of graphs not containing a fixed subgraph and a problem for hypergraphs having no exponent, Graphs and Combinatorics 2 (1986), 113–121.

    Article  MathSciNet  Google Scholar 

  10. P. Erdős, D.J. Kleitman and B.L. Rothschild, Asymptotic enumeration of K n -free graphs, in International Coll. Comb., Atti dei Convegni Lincei (Rome) 17 (1976) 3–17.

    Google Scholar 

  11. P. Erdős and A.H. Stone, On the structure of linear graphs, Bull. Amer. Math. Soc. 52 (1946) 1087–1091.

    Article  MathSciNet  Google Scholar 

  12. D.J. Kleitman and B.L. Rothschild, Asymptotic enumeration of partial orders on a finite set, Trans. Amer. Math. Soc. 205 (1975) (205–220).

    Article  MathSciNet  MATH  Google Scholar 

  13. Ph.G. Kolaitis, H.J. Prömel and B.L. Rothschild, K l+i-free graphs: asymptotic structure and a 0–1-law, Trans. Amer. Math. Soc. 303 (1987) 637–671.

    MathSciNet  MATH  Google Scholar 

  14. H.J. Prömel and A. Steger, Excluding induced subgraphs: quadrilaterals, Random Structures and Algorithms 2 (1991) 55–71.

    Article  MathSciNet  MATH  Google Scholar 

  15. H.J. Prömel and A. Steger, Excluding induced subgraphs II: Extremal graphs Discrete Applied Mathematics (to appear).

    Google Scholar 

  16. H.J. Prömel and A. Steger, Excluding induced subgraphs III: a general asymptotic, Random Structures and Algorithms 3 (1992) 19–31.

    Article  MathSciNet  MATH  Google Scholar 

  17. H.J. Prömel and A. Steger, The asymptotic structure of H-free graphs, in Graph Structure Theory (N. Robertson and P. Seymour, eds), Contemporary Mathematics 147 Amer. Math. Soc., Providence, 1993, pp. 167–178.

    Google Scholar 

  18. H.J. Prömel and A. Steger, Almost all Berge graphs are perfect, Combinatorics, Probability and Computing 1 (1992) 53–79.

    Article  MathSciNet  MATH  Google Scholar 

  19. F.P. Ramsey, On a problem of formal logic, Proc. London Math. Soc. 30(2) (1929) 264–286.

    MATH  Google Scholar 

  20. V. Rödl (personal communication).

    Google Scholar 

  21. V. Rödl, Universality of graphs with uniformly distributed edges, Discrete Mathematics 59 (1986) 125–134.

    Article  MathSciNet  MATH  Google Scholar 

  22. E.R. Scheinerman and J. Zito, On the size of hereditary classes of graphs, J. Combinatorial Theory (B), (to appear).

    Google Scholar 

  23. E. Szémeredi, Regular partitions of graphs, in Proc. Colloque Inter. CNRS (J.-C. Bermond, J.-C. Fournier, M. las Vergnas, D. Sotteau, eds), (1978).

    Google Scholar 

  24. P. Turán, On an extremal problem in graph theory (in Hungarian), Mat. Fiz. Lapok 48 (1941) 436–452.

    MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Pure Mathematics and Mathematical Statistics, 16 Mill Lane, Cambridge, CB2 1SB, England

    Béla Bollobás & Andrew Thomason

Authors
  1. Béla Bollobás
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Andrew Thomason
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. AT&T Bell Laboratories, 600 Mountain Avenue, 07974, Murray Hill, NJ, USA

    Ronald L. Graham

  2. Department of Applied Mathematics, Charles University, Malostranske nam. 25, 11800, Praha, Czech Republic

    Jaroslav Nešetřil

Rights and permissions

Reprints and permissions

Copyright information

© 1997 Springer-Verlag Berlin Heidelberg

About this chapter

Cite this chapter

Bollobás, B., Thomason, A. (1997). Hereditary and Monotone Properties of Graphs. In: Graham, R.L., Nešetřil, J. (eds) The Mathematics of Paul Erdös II. Algorithms and Combinatorics, vol 14. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-60406-5_7

Download citation

  • DOI: https://doi.org/10.1007/978-3-642-60406-5_7

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-64393-4

  • Online ISBN: 978-3-642-60406-5

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation