{
  "abstract": "We give a randomized polynomial-time algorithm that colors any promised $3$-colorable graph on $n$ vertices with $\\smash{O(n^{4/23}) = O(n^{0.17391\\ldots})}$ colors, improving on the recent bounds of $O(n^{0.19539})$ by Bansal, Huang, and Lee and Narang and Tang who obtained $O(n^{(13-\\sqrt{97})/18+ε})=O(n^{0.17506\\dots + ε})$ colors for every fixed $\\smash{ε>0}$. To prove our result, we start from a fixed-level semidefinite relaxation, where we use a finite-depth recursion on Gaussian covers. Fixing a root vertex, we group vertices by correlation with the root vector. Here, each step extends a cover of directions by one edge and transfers it to a successor group. Our key analytic ingredient is a variance bound for Gaussian maxima: for a maximum of $m\\geq 2$ centered linear forms with coefficient norms at most $r$, mean $μ$, and variance $v$, we prove $v\\leq r^2-μ^2/(2\\log m)$ using Chen's Gaussian convexity theorem. Together with a variance-scale lower-tail estimate, this controls the threshold loss at each extension, which shows that root-conditioned vector colorings can either extract a large independent set from a group or bound its size, forcing a contradiction after constantly many steps. The resulting sparse-case guarantee combines with the dense progress bound of Kawarabayashi, Thorup, and Yoneda, and the recursion's numerical inequalities are verified via rational interval arithmetic.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.01071",
  "author_total": 1,
  "authors": [
    {
      "name": "Emile Anand"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/3fxh09v4nxt8t.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.01071",
  "fulltext_url": null,
  "id": "3fxh09v4nxt8t",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.01071",
  "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",
      "cs.CC"
    ],
    [
      "arxiv",
      "cs.DM"
    ],
    [
      "arxiv",
      "cs.DS"
    ],
    [
      "arxiv",
      "math.CO"
    ],
    [
      "openalex",
      "1703|Complexity and Algorithms in Graphs"
    ]
  ],
  "openalex": "W7216409952",
  "pdf_url": "https://arxiv.org/pdf/2610.01071",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/3fxh09v4nxt8t.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:26:07Z",
      "native_id": "2610.01071",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.01071"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Coloring 3-colorable graphs with $O(n^{4/23})$ colors via a Gaussian-cover recursion",
  "title_text": "Coloring 3-colorable graphs with O(n^4/23) colors via a Gaussian-cover recursion",
  "type": "preprint",
  "updated_at": "2026-10-30T12:26:07Z",
  "url": "https://research.draconitic.com/paper/3fxh09v4nxt8t",
  "venue": "arXiv",
  "versions": []
}