Random Graphs

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


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).


Random Graph High Probability Approaches Infinity Theor Comput 
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  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

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