A Sharp Threshold for a Non-monotone Digraph Property

  • Jean-Marie Le Bars
Conference paper
Part of the Trends in Mathematics book series (TM)


We define a non-monotone digraph property TOUR1, a variant of the digraph property KERNEL, which refines the notion of maximal tournament. First we prove that there is a constant 0 < α < 1 such that TOUR1 is asymptotically almost surely true in random digraphs with constant arc probability p ≤ α and asymptotically almost surely false in random digraphs with constant arc probability p > α. Then we concentrate our study on random digraphs with arc probability close to a and we obtain a sharp threshold.


Random Graph Monotone Property Moment Method Small Positive Constant Edge Probability 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    N. Alon and J. Spencer, The probabilistic method. John Wiley ans Sons New York 1992, 254 pp.zbMATHGoogle Scholar
  2. [2]
    B. Bollobas, Random Graphs, Second Edition Cambridge University Press 2001, 498 pp.zbMATHCrossRefGoogle Scholar
  3. [3]
    B. Bollobás and A. Thomason, Threshold Functions. Combinatorica 7, 1987Google Scholar
  4. [4]
    V. Chvãtal. On the computational complexity of finding a kernel. Technical report, Report NO CRM-300, univ. de Montréal, 1973. Centre de Recherches Mathématiques.Google Scholar
  5. [5]
    N. Creignou. The class of problems that are linearly equivalent to satisfiability or a uniform method for proving NP-completeness . Theoretical Computer Science 145:111–145, 1995.MathSciNetzbMATHCrossRefGoogle Scholar
  6. [6]
    P. Erdös and A. Rényi. On the evolution of random graphs. Pupl. Math. Inst. Hungar. Acad. Sci. 7:17–61, 1960.Google Scholar
  7. [7]
    E. Friedgut and G. Kalai. Every monotone graph property has a sharp threshold. Proc. Amer. Math. Soc., 124:2993–3002, 1996MathSciNetzbMATHGoogle Scholar
  8. [8]
    W. Fernandez De La Vega. Kernel on random graphs. Discrete Mathematics 82:213–217, 1990.MathSciNetzbMATHCrossRefGoogle Scholar
  9. [9]
    S. Janson, T. Luczak and A. Rucinski, Random Graphs. John Wiley ans Sons New York 2000, 333 pp.zbMATHCrossRefGoogle Scholar
  10. [10]
    Ph. G. Kolaitis and M.Y. Vardi. 0–1 laws for Fragments of Existential Second-Order Logic: A Survey. Mathematical Foundations of Computer Science pages 84–98, 2000.Google Scholar
  11. [11]
    J-M. Le Bars. Fragments of existential second-order logic without 0–1 laws. In Proceedings of the 13th IEEE Symposium on Logic in Computer Science 1998.Google Scholar
  12. [12]
    J-M. Le Bars. Counterexamples of the 0–1 law for fragments of existential second-order logic: an overview. Bulletin of Symbolic Logic 9:67–82, 2000.CrossRefGoogle Scholar
  13. [13]
    J-M. Le Bars. The 0–1 law fails for frame satisfiability of propositional modal logic. to appear at Symposium on Logic in Computer Science (LICS’2002), 2002. Google Scholar
  14. [14]
    I. Tomescu. Almost all digraphs have a kernel. Discrete Mathematics 84:18–192, 1990.MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer Basel AG 2002

Authors and Affiliations

  • Jean-Marie Le Bars
    • 1
  1. 1.Université de CaenCampus cote de nacreCAENFrance

Personalised recommendations