### Abstract

Let C^{n} 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 C^{n} the n-cube. Let G = G_{n,p} denote the random subgraph of C^{n} defined by letting Prob({i,j} ∈ G) = p for all i, j ∈ C^{n} 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(G_{n,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 language | English (US) |
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Pages (from-to) | 33-39 |

Number of pages | 7 |

Journal | Computers and Mathematics with Applications |

Volume | 5 |

Issue number | 1 |

DOIs | |

State | Published - 1979 |

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### ASJC Scopus subject areas

- Modeling and Simulation
- Computational Theory and Mathematics
- Computational Mathematics

### Cite this

*Computers and Mathematics with Applications*,

*5*(1), 33-39. https://doi.org/10.1016/0898-1221(81)90137-1