Evolution of the n-cube

Paul Erdös, Joel Spencer

Research output: Contribution to journalArticlepeer-review


Let Cn denote the graph with vertices (ε{lunate}1,...,ε{lunate}n), ε{lunate}i = 0, 1 and vertices adjacent if they differ in exactly one coordinate. We call Cn the n-cube. Let G = Gn,p denote the random subgraph of Cn defined by letting Prob({i,j} ∈ G) = p for all i, j ∈ Cn and letting these probabilities be mutually independent. We wish to understand the "evolution" of G as a function of p. Section 1 consists of speculations, without proofs, involving this evolution. Set f{hook}n = Prof(Gn,p is connected) We show in Section 2: Theorem Lim n f{hook}n(p) = 0 if p<0.5e-1 if p=0.51 if p>0.5. The first and last parts were shown by Yu. Burtin[1]. For completeness, we show all three parts.

Original languageEnglish (US)
Pages (from-to)33-39
Number of pages7
JournalComputers and Mathematics with Applications
Issue number1
StatePublished - 1979

ASJC Scopus subject areas

  • Modeling and Simulation
  • Computational Theory and Mathematics
  • Computational Mathematics


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