{
  "abstract": "Brooks' Theorem is a foundational result in graph coloring: if a graph is connected, not a clique, and has maximum degree $Δ\\geq 3$, then it admits a proper $Δ$-coloring. In this paper, we give a new proof of a much stronger result: every such graph admits exponentially many proper $Δ$-colorings. Exponential lower bounds have been obtained previously by applying the Combinatorial Nullstellensatz. In contrast, our proof is via a surprisingly simple and constructive analysis of a textbook greedy coloring algorithm. The entire argument is accessible to non-specialists. We extend our techniques to provide quantitative lower bounds on the number of colorings in several more general settings, namely $Δ$-length list coloring, degree-length list coloring (degree-choosability), and degenerate partitioning. In the latter setting, one is asked to partition the vertices such that each partite set induces a subgraph of bounded degeneracy. In contrast to the other coloring variants considered, no clean Brooks-type characterization is possible, since it is known that for some combinations of parameters determining the existence of even one valid partition is NP-hard. We give a new reduction which significantly extends the parameter regime for which determining the existence of a valid partition is NP-hard. Finally, we give a new proof of the classic degree-choosability theorem, generalizing the techniques behind a recent proof of Brooks' Theorem.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.14374",
  "author_total": 5,
  "authors": [
    {
      "name": "Alex Crane"
    },
    {
      "name": "Carl Feghali"
    },
    {
      "name": "Felicia Lucke"
    },
    {
      "name": "Blair D. Sullivan"
    },
    {
      "name": "Madelyn J. Warr"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/9v2rsyga1h47q.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.14374",
  "fulltext_url": null,
  "id": "9v2rsyga1h47q",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.14374",
  "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": "W7221806188",
  "pdf_url": "https://arxiv.org/pdf/2610.14374",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-09",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/9v2rsyga1h47q.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:15Z",
      "native_id": "2610.14374",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.14374"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Greedy Coloring and Simple Exponential Strengthenings of Brooks-Type Theorems",
  "title_text": "Greedy Coloring and Simple Exponential Strengthenings of Brooks-Type Theorems",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:15Z",
  "url": "https://research.draconitic.com/paper/9v2rsyga1h47q",
  "venue": "arXiv",
  "versions": []
}