### 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) |
---|---|

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

- Applied Mathematics
- Computational Mathematics
- Modeling and Simulation

### Cite this

*Computers and Mathematics with Applications*,

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

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

Research output: Contribution to journal › Article

*Computers and Mathematics with Applications*, vol. 5, no. 1, pp. 33-39. https://doi.org/10.1016/0898-1221(81)90137-1

}

TY - JOUR

T1 - Evolution of the n-cube

AU - Erdös, Paul

AU - Spencer, Joel

PY - 1979

Y1 - 1979

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=0002165630&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0002165630&partnerID=8YFLogxK

U2 - 10.1016/0898-1221(81)90137-1

DO - 10.1016/0898-1221(81)90137-1

M3 - Article

AN - SCOPUS:0002165630

VL - 5

SP - 33

EP - 39

JO - Computers and Mathematics with Applications

JF - Computers and Mathematics with Applications

SN - 0898-1221

IS - 1

ER -