, Volume 2, Issue 1, pp 1–7 | Cite as

Largest random component of ak-cube

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


LetC k denote the graph with vertices (ɛ 1, ...,ɛ k ),ɛ i =0,1 and vertices adjacent if they differ in exactly one coordinate. We callC k thek-cube.

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

We show that forp=λ/k, λ>1,G k, p almost surely contains a connected component of sizec2 k ,c=c(λ). It is also true that the second largest component is of sizeo(2 k ).

AMS subject classification (1980)

05 C 40 60 C 05 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    M. Ajtai, J. Komlós andE. Szemerédi, The longest path in a random graph,Combinatorica 1 (1981) 1–12.zbMATHCrossRefMathSciNetGoogle Scholar
  2. [2]
    P. Erdős andJ. Spencer. Evolution of the n-cube,Computers and Math. with Applications 5 (1979) 33–40.CrossRefGoogle Scholar
  3. [3]
    L. H. Harper, Optimal numberings and isoperimetric problems on graphs,Journal of Comb. Th. 1 (1966) 358–394.MathSciNetGoogle Scholar
  4. [4]
    T. E. Harris,The theory of branching processes, Springer (1963).Google Scholar
  5. [5]
    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

Personalised recommendations