{
  "id": "2311.08560",
  "version": "v1",
  "published": "2023-11-14T21:43:24.000Z",
  "updated": "2023-11-14T21:43:24.000Z",
  "title": "Linear Colouring of Binomial Random Graphs",
  "authors": [
    "Austin Eide",
    "Paweł Prałat"
  ],
  "categories": [
    "math.CO",
    "cs.DM"
  ],
  "abstract": "We investigate the linear chromatic number $\\chi_{\\text{lin}}(G(n,p))$ of the binomial random graph $G(n,p)$ on $n$ vertices in which each edge appears independently with probability $p=p(n)$. For dense random graphs ($np \\to \\infty$ as $n \\to \\infty$), we show that asymptotically almost surely $\\chi_{\\text{lin}}(G(n,p)) \\ge n (1 - O( (np)^{-1/2} ) ) = n(1-o(1))$. Understanding the order of the linear chromatic number for subcritical random graphs ($np < 1$) and critical ones ($np=1$) is relatively easy. However, supercritical sparse random graphs ($np = c$ for some constant $c > 1$) remain to be investigated.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-11-14T21:43:24.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "binomial random graph",
      "linear chromatic number",
      "linear colouring",
      "supercritical sparse random graphs",
      "dense random graphs"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}