### Abstract

Let C^{ k} denote the graph with vertices (e{open}_{ 1}, ..., e{open}_{ k} ), e{open}_{ i} =0,1 and vertices adjacent if they differ in exactly one coordinate. We call C^{ k} the k-cube. Let G=G_{ k, p} denote the random subgraph of C^{ k} defined by letting {Mathematical expression} for all i, j ∈ C^{ k} and letting these probabilities be mutually independent. We show that for p=λ/k, λ>1, G_{ k, p} almost surely contains a connected component of size c2^{ k}, c=c(λ). It is also true that the second largest component is of size o(2^{ k} ).

Original language | English |
---|---|

Pages (from-to) | 1-7 |

Number of pages | 7 |

Journal | Combinatorica |

Volume | 2 |

Issue number | 1 |

DOIs | |

Publication status | Published - Mar 1982 |

### Fingerprint

### Keywords

- AMS subject classification (1980): 05C40, 60C05

### ASJC Scopus subject areas

- Discrete Mathematics and Combinatorics
- Mathematics(all)

### Cite this

*Combinatorica*,

*2*(1), 1-7. https://doi.org/10.1007/BF02579276

**Largest random component of a k-cube.** / Ajtai, M.; Komlós, J.; Szemerédi, E.

Research output: Contribution to journal › Article

*Combinatorica*, vol. 2, no. 1, pp. 1-7. https://doi.org/10.1007/BF02579276

}

TY - JOUR

T1 - Largest random component of a k-cube

AU - Ajtai, M.

AU - Komlós, J.

AU - Szemerédi, E.

PY - 1982/3

Y1 - 1982/3

N2 - Let C k denote the graph with vertices (e{open} 1, ..., e{open} k ), e{open} i =0,1 and vertices adjacent if they differ in exactly one coordinate. We call C k the k-cube. Let G=G k, p denote the random subgraph of C k defined by letting {Mathematical expression} for all i, j ∈ C k and letting these probabilities be mutually independent. We show that for p=λ/k, λ>1, G k, p almost surely contains a connected component of size c2 k, c=c(λ). It is also true that the second largest component is of size o(2 k ).

AB - Let C k denote the graph with vertices (e{open} 1, ..., e{open} k ), e{open} i =0,1 and vertices adjacent if they differ in exactly one coordinate. We call C k the k-cube. Let G=G k, p denote the random subgraph of C k defined by letting {Mathematical expression} for all i, j ∈ C k and letting these probabilities be mutually independent. We show that for p=λ/k, λ>1, G k, p almost surely contains a connected component of size c2 k, c=c(λ). It is also true that the second largest component is of size o(2 k ).

KW - AMS subject classification (1980): 05C40, 60C05

UR - http://www.scopus.com/inward/record.url?scp=51249185094&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=51249185094&partnerID=8YFLogxK

U2 - 10.1007/BF02579276

DO - 10.1007/BF02579276

M3 - Article

AN - SCOPUS:51249185094

VL - 2

SP - 1

EP - 7

JO - Combinatorica

JF - Combinatorica

SN - 0209-9683

IS - 1

ER -