# Evolution of the n-cube

Paul Erdös, Joel Spencer

Research output: Contribution to journalArticle

### Abstract

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. For completeness, we show all three parts.

Original language English (US) 33-39 7 Computers and Mathematics with Applications 5 1 https://doi.org/10.1016/0898-1221(81)90137-1 Published - 1979

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N-cube
Denote
Speculation
Subgraph
Completeness
Graph in graph theory
Theorem

### ASJC Scopus subject areas

• Applied Mathematics
• Computational Mathematics
• Modeling and Simulation

### Cite this

Evolution of the n-cube. / Erdös, Paul; Spencer, Joel.

In: Computers and Mathematics with Applications, Vol. 5, No. 1, 1979, p. 33-39.

Research output: Contribution to journalArticle

Erdös, Paul ; Spencer, Joel. / Evolution of the n-cube. In: Computers and Mathematics with Applications. 1979 ; Vol. 5, No. 1. pp. 33-39.
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