Largest random component of ak-cube

Abstract

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 sizec2k,c=c(λ). It is also true that the second largest component is of sizeo(2k).

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Ajtai, M., Komlós, J. & Szemerédi, E. Largest random component of ak-cube. Combinatorica 2, 1–7 (1982). https://doi.org/10.1007/BF02579276

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AMS subject classification (1980)

  • 05 C 40
  • 60 C 05