{
  "abstract": "Let $ζ(G)$ denote the minimum number of parts in a partition of $V(G)$ in which every part induces either a clique or an independent set. Erdős and Gimbel asked whether, for $G_n\\sim G(n,1/2)$, the difference $χ(G_n)-ζ(G_n)$ tends to infinity with high probability. We resolve this problem along the full sequence $n\\to\\infty$ and prove that $\\mathbb P(χ(G_n)-ζ(G_n)\\ge ((\\log 2)^2/4)\\log(200/153)\\,n/(\\log n)^3)\\to1$. This gives a lower bound at the conjectured scale $n/(\\log n)^3$. We also obtain a phase-resolved refinement: if $δ_n$ is the fractional part of the standard independence-number center, then the coefficient may be replaced by $(\\log 2)^2A_4(δ_n)/4-o(1)$, where $A_4$ is explicit, continuous, nonconstant, and satisfies $A_4(δ)>\\log(200/153)$ for every $δ\\in[0,1]$. The proof uses signed cocoloring profiles supported on four consecutive class sizes and remains uniform across jumps of the natural class-size cutoff. An exact signed-overlap identity separates local cell rewards from a binary cycle-space factor. A canonical decomposition into high cells and a capped residual matching, together with an endpoint-table comparison and an injective restriction of residual even edge sets, yields the required second-moment bound. A bounded-differences argument then amplifies the resulting rare signed witness to a high-probability cocoloring.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.30604",
  "author_total": 1,
  "authors": [
    {
      "name": "Samuil Petkov"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/514s7n7c41faa.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.30604",
  "fulltext_url": null,
  "id": "514s7n7c41faa",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.30604",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.CO"
    ],
    [
      "arxiv",
      "math.PR"
    ],
    [
      "openalex",
      "2607|Limits and Structures in Graph Theory"
    ]
  ],
  "openalex": "W7204961849",
  "pdf_url": "https://arxiv.org/pdf/2608.30604",
  "pmcid": null,
  "pmid": null,
  "published": "2026-08-31",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/514s7n7c41faa.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:17:48Z",
      "native_id": "2608.30604",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.30604"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    }
  ],
  "title": "A Full-Sequence Quantitative Gap Between the Chromatic and Cochromatic Numbers of a Random Graph",
  "title_text": "A Full-Sequence Quantitative Gap Between the Chromatic and Cochromatic Numbers of a Random Graph",
  "type": "preprint",
  "updated_at": "2026-10-30T12:17:48Z",
  "url": "https://research.draconitic.com/paper/514s7n7c41faa",
  "venue": "arXiv",
  "versions": []
}