## Abstract

The binomial random bipartite graph G(n, n, p) is the random graph formed by taking two partition classes of size n and including each edge between them independently with probability p. It is known that this model exhibits a similar phase transition as that of the binomial random graph G(n, p) as p passes the critical point of
^{1}n. We study the component structure of this model near to the critical point. We show that, as with G(n, p), for an appropriate range of p there is a unique ‘giant’ component and we determine asymptotically its order and excess. We also give more precise results for the distribution of the number of components of a fixed order in this range of p. These results rely on new bounds for the number of bipartite graphs with a fixed number of vertices and edges, which we also derive.

Original language | English |
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Article number | P3.7 |

Number of pages | 35 |

Journal | Electronic Journal of Combinatorics |

Volume | 30 |

Issue number | 3 |

DOIs | |

Publication status | Published - 14 Jul 2023 |

## Keywords

- random bipartite graph
- giant component
- excess

## ASJC Scopus subject areas

- Theoretical Computer Science
- Applied Mathematics
- Discrete Mathematics and Combinatorics
- Geometry and Topology
- Computational Theory and Mathematics