The Dimension of Random Graph Orders

  • Béla Bollobás
  • Graham Brightwell
Part of the Algorithms and Combinatorics book series (AC, volume 14)

Summary

The random graph order P n, p is obtained from a random graph G n, p on [n] by treating an edge between vertices i and j, with i ≺ j in [n], as a relation i < j, and taking the transitive closure. This paper forms part of a project to investigate the structure of the random graph order P n, p throughout the range of p = p(n). We give bounds on the dimension of P n, p for various ranges. We prove that, if p log log n → ∞ and ε > 0 then, almost surely,
$$\left( {1 - \in } \right)\sqrt {\frac{{\log n}} {{\log (1/q)}}} \leqslant \dim P_{n,p} \leqslant (1 + \in )\sqrt {\frac{{4\log n}} {{3\log (1/q)}}} .$$
We also prove that there are constants c 1, c 2 such that, if p log n → 0 and p ≥ log n/n, then
$$c_1 p^{ - 1} \leqslant \dim P_{n,p} \leqslant c_2 p^{ - 1} .$$

We give some bounds for various other ranges of p(n), but several questions are left open.

Keywords

Expense milO aCnE 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    M. Albert and A. Frieze, Random graph orders, Order 6 (1989) 19–30.MathSciNetMATHCrossRefGoogle Scholar
  2. 2.
    N. Alon, B. Bollobás, G. Brightwell and S. Janson, Linear extensions of a random partial order, Annals of Applied Prob. 4 (1994) 108–123.MATHCrossRefGoogle Scholar
  3. 3.
    A. Barak and P. Erdős, On the maximal number of strongly independent vertices in a random acyclic directed graph, SIAM J. Algebraic and Disc. Methods 5 (1984) 508–514.MATHCrossRefGoogle Scholar
  4. 4.
    B. Bollobás, Random Graphs, Academic Press, London, 1985, xv+447pp.MATHGoogle Scholar
  5. 5.
    B.Bollobás and G.Brightwell, The width of random graph orders, submitted.Google Scholar
  6. 6.
    B.Bollobás and G.Brightwell, The structure of random graph orders, submitted.Google Scholar
  7. 7.
    B. Bollobás and I. Leader, Isoperimetric inequalities and fractional set systems, J. Combinatorial Theory (A) 56 (1991) 63–74.MATHCrossRefGoogle Scholar
  8. 8.
    G. Brightwell, Models of random partial orders, in Surveys in Combinatorics 1993, Invited papers at the 14th British Combinatorial Conference, K. Walker Ed., Cambridge University Press (1993).Google Scholar
  9. 9.
    P. Erdős, H. Kierstead and W. T. Trotter, The dimension of random ordered sets, Random Structures and Algorithms 2 (1991) 253–275.MathSciNetCrossRefGoogle Scholar
  10. 10.
    P. Erdős and A. Renyi, On random graphs I, Publ. Math. Debrecen 6 (1959) 290–297.MathSciNetGoogle Scholar
  11. 11.
    P. Erdős and A. Renyi, On the evolution of random graphs, Publ. Math. Inst. Hungar. Acad. Sci. 5 (1960) 17–61.Google Scholar
  12. 12.
    Z. Fiiredi and J. Kahn, On the dimensions of ordered sets of bounded degree, Order 3 (1986) 17–20.Google Scholar
  13. 13.
    M. Hall, Combinatorial Theory, 2nd Edn., Wiley-Interscience Series in Discrete Mathematics (1986) xv+440pp.Google Scholar
  14. 14.
    G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 5th Edn., Oxford University Press (1979) xvi+426pp.MATHGoogle Scholar
  15. 15.
    I. Leader, Discrete isoperimetric inequalities, in Probabilistic Combinatorics and its Applications, Proceedings of Symposia in Applied Mathematics 44, American Mathematical Society, Providence (1991).Google Scholar
  16. 16.
    C. M. Newman, Chain lengths in certain directed graphs, Random Structures and Algorithms 3 (1992) 243–253.MathSciNetMATHCrossRefGoogle Scholar
  17. 17.
    C. M. Newman and J. E. Cohen, A stochastic theory of community food webs: IV. Theory of food chain lengths in large webs, Proc. R. Soc. London Ser. B 228 (1986) 355–377.CrossRefGoogle Scholar
  18. 18.
    K. Simon, D. Crippa and F. Collenberg, On the distribution of the transitive closure in random acyclic digraphs, Lecture Notes in Computer Science 726 (1993) 345–356.Google Scholar
  19. 19.
    W. T. Trotter, Inequalities in dimension theory for posets, Proc. Amer. Math. Soc. 47 (1975) 311–316.MathSciNetMATHCrossRefGoogle Scholar
  20. 20.
    W. T. Trotter, Embedding finite posets in cubes, Discrete Math. 12 (1975) 165–172.MathSciNetMATHCrossRefGoogle Scholar
  21. 21.
    W. T. Trotter, Combinatorics and Partially Ordered Sets: Dimension Theory, The Johns Hopkins University Press, Baltimore (1992) xiv+307ppMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1997

Authors and Affiliations

  • Béla Bollobás
    • 1
  • Graham Brightwell
    • 2
  1. 1.Department of Pure Mathematics and Statistics of MathematicsUniversity of CambridgeCambridgeUK
  2. 2.Department of Pure MathematicsLondon School of Economics and Political ScienceLondonUK

Personalised recommendations