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 sizec2^k,c=c(λ). It is also true that the second largest component is of sizeo(2^k).