, Volume 2, Issue 1, pp 1–7 | Cite as

Largest random component of ak-cube

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


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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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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