Advertisement

Random Graphs

  • Xueliang Li
  • Colton Magnant
  • Zhongmei Qin
Chapter
  • 326 Downloads
Part of the SpringerBriefs in Mathematics book series (BRIEFSMATH)

Abstract

For random graphs, the following results were shown in Gu et al. (Theor Comput Sci 609:336–343, 2016). Here let G(n, p) denote the Erdős-Renyi (Erdős and Rényi, Magy Tud Akad Mat Kutató Int Közl 5:17–61, 1960) random graph with n vertices and edges appearing with probability p. We say an event \(\mathcal {A}\) happens with high probability if the probability that it happens approaches 1 as n →, i.e., \(Pr[\mathcal {A}]=1-o_n(1)\). Sometimes, we say w.h.p. for short. We say that a property holds for almost all graphs if the probability of the property holding for G(n, 1∕2) approaches 1 as n approaches infinity. The first result follows easily from Theorem  3.0.2 and the fact that almost all graphs are 3-connected (Blass and Harary, J Graph Theory 3:225–240, 1979).

Keywords

Random Graph High Probability Approaches Infinity Theor Comput 
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.

References

  1. 8.
    Blass, A., Harary, F.: Properties of almost all graphs and complexes. J. Graph Theory 3(3), 225–240 (1979)MathSciNetCrossRefGoogle Scholar
  2. 9.
    Bollobás, B.: Random Graphs. Cambridge University Press, Cambridge (2001)Google Scholar
  3. 24.
    Cooper, C., Frieze, A.: Pancyclic random graphs. Proc. Conf. Random Graphs, Poznán (1987)Google Scholar
  4. 30.
    Erdös, P., Rényi, A.: On the evolution of random graphs. Publ. Math. Inst. Hungar. Acad. Sci. 5, 17–61 (1960)MathSciNetzbMATHGoogle Scholar
  5. 31.
    Frieze, A., Krivelevich, M.: On two Hamilton cycle problems in random graphs. Isr. J. Math. 166, 221–234 (2008)MathSciNetCrossRefGoogle Scholar
  6. 34.
    Gu, R., Li, X., Qin, Z.: Proper connection number of random graphs. Theor. Comput. Sci. 609, 336–343 (2016)MathSciNetCrossRefGoogle Scholar
  7. 64.
    Pósa, L.: Hamiltonian circuits in random graphs. Discrete Math. 14, 359–364 (1976)MathSciNetCrossRefGoogle Scholar

Copyright information

© The Author(s), under exclusive licence to Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  • Xueliang Li
    • 1
  • Colton Magnant
    • 2
  • Zhongmei Qin
    • 3
  1. 1.Center for CombinatoricsNankai UniversityTianjinChina
  2. 2.Department of MathematicsGeorgia Southern UniversityStatesboroUSA
  3. 3.College of ScienceChang’an UniversityXi’anChina

Personalised recommendations