, Volume 2, Issue 1, pp 1-7

Largest random component of ak-cube

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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 ).