{
  "id": "2212.07880",
  "version": "v1",
  "published": "2022-12-15T15:04:40.000Z",
  "updated": "2022-12-15T15:04:40.000Z",
  "title": "Twin-width of random graphs",
  "authors": [
    "Jungho Ahn",
    "Debsoumya Chakraborti",
    "Kevin Hendrey",
    "Donggyu Kim",
    "Sang-il Oum"
  ],
  "comment": "34 pages, 3 figures",
  "categories": [
    "math.CO",
    "cs.DM"
  ],
  "abstract": "We investigate the twin-width of the Erd\\H{o}s-R\\'enyi random graph $G(n,p)$. We unveil a surprising behavior of this parameter by showing the existence of a constant $p^*\\approx 0.4$ such that with high probability, when $p^*\\le p\\le 1-p^*$, the twin-width is asymptotically $2p(1-p)n$, whereas, when $p<p^*$ or $p>1-p^*$, the twin-width is significantly higher than $2p(1-p)n$. In particular, we show that the twin-width of $G(n,1/2)$ is concentrated around $n/2 - (\\sqrt{3n \\log n})/2$ within an interval of length $o(\\sqrt{n\\log n})$. For the sparse random graph, we show that with high probability, the twin-width of $G(n,p)$ is $\\Theta(n\\sqrt{p})$ when $(726\\ln n)/n\\leq p\\leq1/2$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-12-15T15:04:40.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C35"
    ],
    "keywords": [
      "twin-width",
      "high probability",
      "sparse random graph",
      "surprising behavior"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 34,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}