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Random graph orders

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Abstract

Let P n be the order determined by taking a random graph G on {1, 2,..., n}, directing the edges from the lesser vertex to the greater (as integers), and then taking the transitive closure of this relation. We call such and ordered set a random graph order. We show that there exist constants c, and °, such that the expected height and set up number of P n are sharply concentrated around cn and °n respectively. We obtain the estimates: .565<c<.610, and .034<°<.289. We also discuss the width, dimension, and first-order properties of P n.

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References

  1. K. Azuma (1967) Weighted sums of certain dependent random variables, Tokuku Math. J. 19, 357–367.

    Google Scholar 

  2. A. Barak and P. Erdös (1984) On the maximal number of strongly independent vertices in a random acyclic directed graph, SIAM J. Algebraic Discete Methods 5, 508–514.

    Google Scholar 

  3. K. Compton (1987) The computational complexity of asymptotic problems I: Partial orders (preprint).

  4. R. Fagin (1976) Probabilities on finite models, J. Symbolic Logic 41, 50–58.

    Google Scholar 

  5. William Feller (1950) An Introduction to Probability Theory and its Applications, (2nd edn.), Wiley, New York.

    Google Scholar 

  6. G. Gierz and W. Poguntke (1983) Minimizing setups for ordered sets: a linear algebraic approach, SIAM J. Algebraic Discrete Methods 4, 132–144.

    Google Scholar 

  7. W. Hoeffding (1963) Probability inequalities of sums of bounded random variables, J. Amer. Stat. Assoc. 58, 13–30.

    Google Scholar 

  8. S. Karlin and H. M. Taylor (1975) A First Course in Stochastic Processes. Academic Press, New York.

    Google Scholar 

  9. G. Polya and G. Szëgo (1972) Problems and Theorems in Analysis 1, Springer-Verlag, New York.

    Google Scholar 

  10. E. Shamir and J. Spencer (1987) Sharp concentration of the chromatic numbers on random graphs G n,p, Combinatiorica 7, 121–129.

    Google Scholar 

  11. P. Winkler (1984) Random orders, Order 1, 317–331.

    Google Scholar 

  12. P. Winkler (1985) Connectedness and diameter for random orders of fixed dimension. Order 2, 165–171.

    Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, Carnegie Mellon University, 15213, Pittsburgh, PA, U.S.A.

    Michael H. Albert & Alan M. Frieze

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  1. Michael H. Albert
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  2. Alan M. Frieze
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Communicated by P. Hell

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Albert, M.H., Frieze, A.M. Random graph orders. Order 6, 19–30 (1989). https://doi.org/10.1007/BF00341633

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  • DOI: https://doi.org/10.1007/BF00341633

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