Combinatorica

, Volume 2, Issue 1, pp 1–7 | Cite as

Largest random component of ak-cube

  • M. Ajtai
  • J. Komlós
  • E. Szemerédi
Article

Abstract

LetCk denote the graph with vertices (ɛ1, ...,ɛk),ɛi=0,1 and vertices adjacent if they differ in exactly one coordinate. We callCk thek-cube.

LetG=Gk, p denote the random subgraph ofCk defined by letting
$$Prob(\{ i,j\} \in G) = p$$
for alli, j ∈ Ck and letting these probabilities be mutually independent.

We show that forp=λ/k, λ>1,Gk, p almost surely contains a connected component of sizec2k,c=c(λ). It is also true that the second largest component is of sizeo(2k).

AMS subject classification (1980)

05 C 40 60 C 05 

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References

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    M. Ajtai, J. Komlós andE. Szemerédi, The longest path in a random graph,Combinatorica 1 (1981) 1–12.MATHCrossRefMathSciNetGoogle Scholar
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    P. Erdős andJ. Spencer. Evolution of the n-cube,Computers and Math. with Applications 5 (1979) 33–40.CrossRefGoogle Scholar
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    L. H. Harper, Optimal numberings and isoperimetric problems on graphs,Journal of Comb. Th. 1 (1966) 358–394.MathSciNetGoogle Scholar
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    T. E. Harris,The theory of branching processes, Springer (1963).Google Scholar
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    J. Komlós, M. Sulyok andE. Szemerédi. Underdogs in a random graph,submitted to Studia Sci. Math. Hung. Google Scholar

Copyright information

© Akadémiai Kiadó 1982

Authors and Affiliations

  • M. Ajtai
    • 1
  • J. Komlós
    • 1
  • E. Szemerédi
    • 1
  1. 1.Mathematical Institute of the Hungarian Academy of SciencesBudapestHungary

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