### 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 |
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Pages (from-to) | 1-7 |

Number of pages | 7 |

Journal | Combinatorica |

Volume | 2 |

Issue number | 1 |

DOIs | |

Publication status | Published - Mar 1 1982 |

### Keywords

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

### ASJC Scopus subject areas

- Discrete Mathematics and Combinatorics
- Computational Mathematics

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## Cite this

Ajtai, M., Komlós, J., & Szemerédi, E. (1982). Largest random component of a k-cube.

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