{
  "abstract": "It was recently shown that every connected graph of order $n \\geq 5$ and size $m$ satisfies $χ(G) \\leq \\left\\lceil \\frac{m}{\\sqrt{n}} \\right\\rceil$, and it was asked whether the stronger inequality $χ(G) \\leq \\left\\lceil \\frac{m}{2\\sqrt{n}} \\right\\rceil + 1$ holds for every connected triangle-free graph. In this paper, we answer this question in the affirmative. In fact, we prove that every connected triangle-free graph $G$ with $G \\not\\cong C_5$ satisfies $χ(G) \\leq \\left\\lceil \\frac{m}{\\sqrt{5.5n}} \\right\\rceil + 1$, where the constant $\\sqrt{5.5}$ cannot be replaced by any constant greater than or equal to $\\sqrt{6}$, and the equality holds for the Grötzsch graph and for every odd cycle of length between $7$ and $21$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.07014",
  "author_total": 2,
  "authors": [
    {
      "name": "Saieed Akbari"
    },
    {
      "name": "Arash Beikmohammadi"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/46vp838gvsgje.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.07014",
  "fulltext_url": null,
  "id": "46vp838gvsgje",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.07014",
  "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"
    ],
    [
      "openalex",
      "1703|Advanced Graph Theory Research"
    ]
  ],
  "openalex": "W7211996165",
  "pdf_url": "https://arxiv.org/pdf/2609.07014",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-07",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/46vp838gvsgje.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:19:58Z",
      "native_id": "2609.07014",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.07014"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Bounds for the Vertex Chromatic Number of Connected Triangle-Free Graphs",
  "title_text": "Bounds for the Vertex Chromatic Number of Connected Triangle-Free Graphs",
  "type": "preprint",
  "updated_at": "2026-10-30T12:19:58Z",
  "url": "https://research.draconitic.com/paper/46vp838gvsgje",
  "venue": "arXiv",
  "versions": []
}