Random interval graphs

Abstract

In this paper we introduce a notion ofrandom interval graphs: the intersection graphs of real, compact intervals whose end points are chosen at random. We establish results about the number of edges, degrees, Hamiltonicity, chromatic number and independence number of almost all interval graphs.

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

References

  1. [1]

    P. Billingsley, Probability and Measure, Wiley, New York (1979).

    Google Scholar 

  2. [2]

    B. Bollobás, Random Graphs, Academic Press, New York (1985).

    Google Scholar 

  3. [3]

    V. Chvatal, On Hamiltonian ideals, J. Comb. Theory B12 (1972), 163–168.

    Article  Google Scholar 

  4. [4]

    J. E. Cohen, The asymptotic probability that a random graph is a unit interval graph, indifference graph or proper interval graph, Discrete Math.40 (1982), 21–24.

    Article  Google Scholar 

  5. [5]

    P. Erdős andA. Rényi, On random graphs I, Publ. Math. Debrecen6 (1959), 290–297.

    Google Scholar 

  6. [6]

    P. Erdős andA. Rényi, On the evolution of random graphs, Magyar Tud. Akad. Mat. Kutató Int. Közl.5 (1960), 17–61.

    Google Scholar 

  7. [7]

    P. Erdős andD. West, A note on the interval number of a graph, Discrete Math.55 (1985), 129–133.

    Article  Google Scholar 

  8. [8]

    P. Gilmore andA. Hoffman, A characterization of comparability graphs and of interval graphs, Canad. J. Math.16 (1964), 539–548.

    Google Scholar 

  9. [9]

    M. Golumbic, Algorithmic Graphs Theory and Perfect Graphs, Academic Press, New York (1980).

    Google Scholar 

  10. [10]

    E. Helly, Uber Mengen Kurper mit gemeinschaftlichen Punkten, J. Deutsch Math. Verein32 (1923), 175–176.

    Google Scholar 

  11. [11]

    C. Lekkerkerker andJ. Boland, Representation of a finite graph by a set of intervals on the real line, Fund. Math.51 (1974), 45–64.

    Google Scholar 

  12. [12]

    E. Palmer, Graphical Evolution: An Introduction to the Theory of Random Graphs, Wiley, New York (1985).

    Google Scholar 

  13. [13]

    Lord, Rayleigh, On the resultant of a large number of vibrations of the same pitch and of arbitrary phase, Philosophical Magazine and Journal of Science10 (1880), 73–78.

    Google Scholar 

Download references

Author information

Affiliations

Authors

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Scheinerman, E.R. Random interval graphs. Combinatorica 8, 357–371 (1988). https://doi.org/10.1007/BF02189092

Download citation

AMS subject classification (1980)

  • 05 C 30