Skip to main content
SpringerLink
Log in

The Origins of the Theory of Random Graphs

  • Chapter
The Mathematics of Paul Erdös I

Part of the book series: Algorithms and Combinatorics ((AC,volume 13))

Abstract

The origins of the theory of random graphs are easy to pin down. Undoubtfully one should look at a sequence of eight papers co-authored by two great mathematicians: Paul Erdős and Alfred Renyi, published between 1959 and 1968:

  1. [ER59]

    On random graphs I, Publ. Math. Debrecen 6 (1959), 290–297.

  2. [ER60]

    On the evolution of random graphs, Publ. Math. Inst. Hung. Acad. Sci. 5 (1960), 17–61.

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

Access this chapter

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

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. M. Ajtai, J. Komlós and E. Szemerédi Topological complete subgraphs in random graphs, Studia. Sci. Math. Hungar. 14 (1979), 293–297.

    MathSciNet  MATH  Google Scholar 

  2. N. Alon and J. Spencer, The Probabilistic Method, Wiley, 1992.

    MATH  Google Scholar 

  3. N. Alon and R. Yuster, Threshold functions for H-factors, Combinatorics, Probability and Computing (1993).

    Google Scholar 

  4. A. D. Barbour, Poisson convergence and random-graphs. Math. Proc. Cambr. Phil. Soc. 92 (1982), 349–359.

    Article  MathSciNet  MATH  Google Scholar 

  5. A. D. Barbour, S. Janson, M. Karoński and A. Ruciński. Small cliques in random graphs, Random Structures Alg. 1 (1990), 403–434.

    Article  MATH  Google Scholar 

  6. A. D. Barbour. M. Karoński and A. Ruciński, A central limit theorem for decomposable random variables with applications to random graphs. J. Comb. Th.-B 47 (1989), 125–145.

    Article  MATH  Google Scholar 

  7. P. Billingsley, Probability and Measure, Wiley, 1979.

    MATH  Google Scholar 

  8. B. Bollobas, Threshold functions for small subgraphs, Math. Proc. Cambr. Phil. Soc. 90 (1981), 197–206.

    Article  MathSciNet  MATH  Google Scholar 

  9. B. Bollobas, Vertices of given degree in a random graph, J. Graph Theory 6 (1982), 147–155.

    Article  MathSciNet  MATH  Google Scholar 

  10. B. Bollobas, Distinguishing vertices of random graphs, Annals Discrete Math. 13 (1982), 33–50.

    MATH  Google Scholar 

  11. B. Bollobas Almost all regular graphs are Hamiltonian, Europ. J. Combinatorics 4 (1983), 97–106.

    MathSciNet  MATH  Google Scholar 

  12. B. Bollobas, The evolution of random graphs Trans. Amer. Math. Soc. 286 (1984), 257–274.

    Article  MathSciNet  MATH  Google Scholar 

  13. B. Bollobas, The chromatic number of random graphs, Combinatorica 8 (1988) 49–55.

    Article  MathSciNet  MATH  Google Scholar 

  14. B. Bollobas and P. Erdős, Cliques in random graphs, Math. Proc. Cambr. Phil. Soc. 80 (1976), 419–427.

    Article  MATH  Google Scholar 

  15. B. Bollobas, Random Graphs, Academic Press, London, 1985.

    MATH  Google Scholar 

  16. B. Bollobas and A. Frieze, On matchings and hamiltonian cycles in random graphs, Random Graphs’83, vol. 28, Annals of Discrete Mathematics, 1985, pp. 1–5.

    Google Scholar 

  17. B. Bollobas and A. Thomason, Random graphs of small order, Annals of Discrete Math. 28 (1985), 47–98.

    MathSciNet  Google Scholar 

  18. B. Bollobas and A. Thomason, Threshold functions, Combinatorica 7 (1987), 35–38.

    Article  MathSciNet  MATH  Google Scholar 

  19. B. Bollobas and J.C. Wierman, Subgraph counts and containment probabilities of balanced and unbalanced subgraphs in a large random graph, Graph Theory and Its Applications: East and West (Proc. 1st China-USA Intern. Graph Theory Conf.), Eds. Capobianco et al. vol. 576, Annals of the New York Academy of Sciences, 1989 pp. 63–70.

    Google Scholar 

  20. P. Erdős, Some remarks on the theory of graphs, Bull. Ainer. Math. Soc. 53 (1947), 292–294.

    Article  Google Scholar 

  21. P. Erdős and A. Renyi, On the evolution of random graphs, Bull. Inst. Internat. Statist. 38 (1961), 343–347.

    MathSciNet  Google Scholar 

  22. P. Erdős and A. Renyi, On random graphs I, Publ. Math. Debrecen 6 (1959), 290–297.

    MathSciNet  Google Scholar 

  23. P. Erdős and A. Renyi, On the evolution of random graphs, Publ. Math. Inst. Hung. Acad. Sci. 5 (1960), 17–61.

    Google Scholar 

  24. P. Erdős and A. Renyi, On the strength of connectedness of a random graph, Acta Math. Acad. Sci. Hungar. 12 (1961), 261–267.

    Article  MathSciNet  Google Scholar 

  25. P. Erdős and A. Renyi, On random matrices, Publ. Math. Inst. Hung. Acad. Sci. 8 (1964), 455–461.

    Google Scholar 

  26. P. Erdős and A. Renyi, On the existence of a factor of degree one of a connected random graph, Acta Math. Acad. Sci. Hung. 17 (1966), 359–368.

    Article  Google Scholar 

  27. P. Erdős and A. Renyi, Asymmetric graphs, Acta Math. Acad. Sci. Hung. 14 (1963), 295–315.

    Article  Google Scholar 

  28. P. Erdős and A. Renyi, On random matrices, II, Studia Sci. Math. Hung. 3 (1968), 459–464.

    Google Scholar 

  29. A. Frieze and S. Janson, submitted.

    Google Scholar 

  30. E. N. Gilbert, Random graphs, Annals of Mathematical Statistics 30 (1959), 1141–1144.

    Article  MathSciNet  MATH  Google Scholar 

  31. E. Godehardt and J. Steinbach, On a lemma of P. Erdős and A. Renyi about random graphs, Publ. Math. 28 (1981), 271–273.

    MATH  Google Scholar 

  32. G. R. Grimmett and C. J. H. McDiarmid, On colouring random graphs. Math. Proc. Cambr. Phil. Soc. 77 (1975), 313–324.

    Article  MathSciNet  MATH  Google Scholar 

  33. S. Janson, D. E. Knuth, T. Luczak and B. Pittel, The birth of the giant component, Random Structures & Algorithms 4 (1993), 233–358.

    Article  MathSciNet  MATH  Google Scholar 

  34. S. Janson and J. Kratochvil, Proportional graphs, Random Structures Algorithms 2 (1991) 209–224.

    Article  MathSciNet  MATH  Google Scholar 

  35. S. Janson, T. Luczak and A. Ruciński, An exponential bound for the probability of nonexistence of a specified subgraph of a random graph, Proceedings of Random Graphs’ 87, Wiley, Chichester, 1990, pp. 73–87.

    Google Scholar 

  36. S. Janson and J. Spencer, Probabilistic constructions of proportional graphs, Random Structures Algorithms 3 (1992), 127–137.

    Article  MathSciNet  MATH  Google Scholar 

  37. M. Karohski, On the number of k-trees in a random graph, Prob. Math. Stat. 2 (1982) 197–205.

    Google Scholar 

  38. M. Karoński and A. Ruciński, On the number of strictly balanced subgraphs of a random graph, Graph Theory, Lagow 1981, Lecture Notes in Math. 1018, Springer-Verlag 1983, 79–83.

    Google Scholar 

  39. J. Karrman, Existence of proportional graphs, J. Graph Theory 17 (1993) 207–220.

    Article  MathSciNet  Google Scholar 

  40. V. F. Kolchin, On the limit behavior of a random graph near the critical point Theory Probability Its Appl. 31 (1986), 439–451.

    Article  MathSciNet  Google Scholar 

  41. J. Komlós and E. Szemerédi, Limit distributions for the existence of Hamilton cycles, Discrete Math. 43 (1983), 55–63.

    Article  MathSciNet  MATH  Google Scholar 

  42. A. D. Korshunov, A solution of a problem of Erdős and Renyi on Hamilton cycles in non-oriented graphs, Metody Diskr. Anal. 31 (1977), 17–56.

    MATH  Google Scholar 

  43. T. Luczak, The automorphism group of random graphs with a given number of edges, Math. Proc. Camb. Phil. Soc. 104 (1988), 441–449.

    Article  MathSciNet  MATH  Google Scholar 

  44. T. Luczak, On the chromatic number of sparse random graphs, Combinatorica 10 (1990), 377–385.

    Google Scholar 

  45. T. Luczak, Component behavior near the critical point of the random graph process, Random Structures & Algorithms 1 (1990), 287–310.

    Article  MathSciNet  MATH  Google Scholar 

  46. T. Luczak, On the equivalence of two basic models of random graphs, Proceedings of Random Graphs ’87, Wiley, Chichester, 1990, pp. 73–87.

    Google Scholar 

  47. T. Luczak, A note on the sharp concentration of the chromatic number of a random graph, Combinatorica 11 (1991), 295–297.

    Article  MathSciNet  MATH  Google Scholar 

  48. T. Luczak and J.C. Wierman, The chromatic number of random graphs at the double-jump threshold, Combinatorica 9 (1989), 39–49.

    Article  MathSciNet  MATH  Google Scholar 

  49. T. Luczak, B. Pittel and J.C. Wierman, The structure of a random graph at the double-jump threshold, Trans. Am. Math. Soc. 341 (1994), 721–728.

    Article  MathSciNet  MATH  Google Scholar 

  50. T. Luczak, A. Ruciński, Tree-matchings in random graph processes, SIAM J. Discr. Math 4 (1991) 107–120.

    Article  MATH  Google Scholar 

  51. D. Matula, The employee party problem, Notices Amer. Math. Soc. 19 (1972), A-382.

    Google Scholar 

  52. D. W. Matula, The largest clique size in a random graph, Tech. Rep. Dept. Comput. Sci. Southern Methodist Univ., Dallas (1976).

    Google Scholar 

  53. D. Matula and L. Kucera, An expose-and-merge algorithm and the chromatic number of a random graph, Random Graphs ’87 J. Jaworski, M. Karoński and A. Ruciński, eds.), John Wiley & Sons, New York, 1990, 175–188.

    Google Scholar 

  54. L. Posa, Hamiltonian circuits in random graphs, Discrete Math. 14 (1976), 359–364.

    Article  MathSciNet  MATH  Google Scholar 

  55. A. Ruciński, When are small subgraphs of a random graph normally distributed?, Prob. Th. Rel. Fields 78 (1988) 1–10.

    Article  MATH  Google Scholar 

  56. A. Ruciński, Small subgraphs of random graphs: a survey, Proceedings of Random Graphs ’87, Wiley, Chichester, 1990, 283–303.

    Google Scholar 

  57. A. Ruciński, Matching and covering the vertices of a random graph by copies of a given graph, Discrete Math. 105 (1992), 185–197.

    Article  MathSciNet  MATH  Google Scholar 

  58. A. Ruciński and A. Vince, Balanced graphs and the problem of subgraphs of random graphs, Congres. Numerantium 49 (1985), 181–190.

    Google Scholar 

  59. K. Schü;rger, Limit theorems for complete subgraphs of random graphs, Period. Math. Hungar. 10 (1979), 47–53.

    Article  MathSciNet  Google Scholar 

  60. J. Schmidt and E. Shamir, A threshold for perfect matchings in random d-pure hypergraphs, Discrete Math.45 (1983) 287–295.

    Article  MathSciNet  MATH  Google Scholar 

  61. E. Shamir and J. Spencer, Sharp concentration of the chromatic number of random graphs G n , P , Combinatorica 7 (1987), 121–129.

    Article  MathSciNet  MATH  Google Scholar 

  62. E. Shamir and E. Upfal, On factors in random graphs, Israel J. Math. 39 (1981), 296–302.

    Article  MathSciNet  MATH  Google Scholar 

  63. V. E. Stepanov, On some features of the structure of a random graph near the critical point, Theory Prob. Its Appl. 32 (1988), 573–594.

    Article  Google Scholar 

  64. E. M. Wright, Asymmetric and symmetric graphs, Glasgow Math. J. 15 (1974), 69–73.

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Faculty of Mathematics and Computer Science, Adam Mickiewicz University, Poznań, Poland

    Micha Karoński & Andrzej Ruciński

  2. Department of Mathematics and Computer Science, Emory University, Atlanta, Georgia, USA

    Micha Karoński & Andrzej Ruciński

Authors
  1. Micha Karoński
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Andrzej Ruciński
    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, NJ 07974, Murray Hill, 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

Karoński, M., Ruciński, A. (1997). The Origins of the Theory of Random Graphs. In: Graham, R.L., Nešetřil, J. (eds) The Mathematics of Paul Erdös I. Algorithms and Combinatorics, vol 13. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-60408-9_24

Download citation

  • DOI: https://doi.org/10.1007/978-3-642-60408-9_24

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-64394-1

  • Online ISBN: 978-3-642-60408-9

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation