{
  "abstract": "An equitable $k$-coloring of a graph partitions its vertex set into $k$ independent sets whose sizes differ by at most one; the least such $k$ is the equitable chromatic number $\\chie(G)$. Every known bound on $\\chie$ valid for all graphs, beginning with the Hajnal--Szemerédi theorem, is linear in the maximum degree $Δ$, and the star $K_{1,Δ}$, for which $\\chie=\\ceil{Δ/2}+1$, shows that no general bound below $Δ/2$ exists. We prove that this obstruction is a shortage of vertices rather than an effect of the degree: every graph with $|V(G)|\\ge3χ(G)Δ$ satisfies $\\chie(G)=O\\bigl(χ(G)^{3/2}\\sqrt{Δ/\\lnΔ}\\bigr)$ throughout the range $χ(G)\\le(Δ/\\lnΔ)^{1/3}$, so that for graphs of large order the degree enters only through $\\sqrt{Δ/\\lnΔ}$, with the chromatic number governing the rest. For each fixed $\\ell$, $\\ell$-colorable graphs of sufficiently large order satisfy $\\chie\\le\\bigl(2\\sqrt2\\,\\ell\\sqrt{\\ell-1}+o(1)\\bigr)\\sqrt{Δ/\\lnΔ}$, while a probabilistic construction supplies graphs of arbitrarily large order, bipartite when $\\ell=2$, with $\\chie\\ge\\tfrac13\\sqrt{(\\ell-1)Δ/\\lnΔ}$: the order of growth $Θ\\bigl(\\sqrt{Δ/\\lnΔ}\\bigr)$ is exact for every fixed chromatic number, and the extremal constant is determined up to a factor $O(χ(G))$. All upper bounds are constructive, and a prescribed-anchor variant of the construction produces equitable colorings of bipartite graphs with $O(\\sqrtΔ)$ colors in optimal linear time.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.25041",
  "author_total": 1,
  "authors": [
    {
      "name": "Mohammad F. Marashdeh"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/ac0d6rpndvrr1.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.25041",
  "fulltext_url": null,
  "id": "ac0d6rpndvrr1",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.25041",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.CO"
    ],
    [
      "openalex",
      "1703|Advanced Graph Theory Research"
    ]
  ],
  "openalex": "W7214101458",
  "pdf_url": "https://arxiv.org/pdf/2609.25041",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/ac0d6rpndvrr1.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:23:43Z",
      "native_id": "2609.25041",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.25041"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "A square-root law for equitable coloring",
  "title_text": "A square-root law for equitable coloring",
  "type": "preprint",
  "updated_at": "2026-10-30T12:23:43Z",
  "url": "https://research.draconitic.com/paper/ac0d6rpndvrr1",
  "venue": "arXiv",
  "versions": []
}