{
  "id": "2105.14883",
  "version": "v1",
  "published": "2021-05-31T11:12:02.000Z",
  "updated": "2021-05-31T11:12:02.000Z",
  "title": "Component behaviour and excess of random bipartite graphs near the critical point",
  "authors": [
    "Tuan Anh Do",
    "Joshua Erde",
    "Mihyun Kang"
  ],
  "comment": "35 pages",
  "categories": [
    "math.CO"
  ],
  "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 $\\frac{1}{n}$. 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.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-05-31T11:12:02.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "critical point",
      "component behaviour",
      "binomial random bipartite graph",
      "binomial random graph",
      "similar phase transition"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 35,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}