{
  "abstract": "For an \\(n\\)-vertex graph \\(G\\) and an integer \\(m\\ge n\\), let \\(f'(G,m)\\) be the minimum number of distinct color sets used by proper colorings from \\(m\\)-element lists that are disjoint at distinct nonadjacent vertices. A clique-partition assignment gives a common list to the vertices of each part of a clique partition and uses disjoint lists on different parts. In this paper, we construct a graph on \\(20\\) vertices for which these assignments do not attain \\(f'(G,20)\\), and a fixed graph on \\(25\\) vertices for which they fail to attain the minimum for every allowed list size. For every triangle-free graph \\(G\\) on \\(n\\ge2\\) vertices, we prove \\[ f'(G,m)=\\binom{m}{2}^{ν(G)}m^{n-2ν(G)} \\qquad (m\\ge n), \\] where \\(ν(G)\\) is the size of a maximum matching, and characterize all minimizing assignments in terms of maximum matchings. For clique blow-ups of each fixed odd cycle of length at least five, we determine the minimum up to constant factors uniformly over the blow-up size and all allowed list sizes, and obtain an exponential separation from the clique-partition value. We also prove that a noncomplete graph \\(G\\) satisfies \\(f'(G,n)=n^2\\) if and only if all edges of its complement have a common endpoint. These results disprove the general clique-partition conjecture of Gravier, Maffray, and Mohar (\\emph{Discrete Mathematics}, 2003) and its extension by Wu and Zhang (\\emph{Applied Mathematics Letters}, 2006), while establishing both triangle-free conjectures.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2611.07924",
  "author_total": 1,
  "authors": [
    {
      "name": "Jun Yue"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/080aeqs2xw74m.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2611.07924",
  "fulltext_url": null,
  "id": "080aeqs2xw74m",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2611.07924",
  "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": null,
  "pdf_url": "https://arxiv.org/pdf/2611.07924",
  "pmcid": null,
  "pmid": null,
  "published": "2026-11-05",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/080aeqs2xw74m.ris",
  "sources": [
    {
      "first_seen": "2026-11-06T18:21:34Z",
      "native_id": "2611.07924",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2611.07924"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Clique partitions and distinct color sets in list colorings",
  "title_text": "Clique partitions and distinct color sets in list colorings",
  "type": "preprint",
  "updated_at": "2026-11-06T18:21:34Z",
  "url": "https://research.draconitic.com/paper/080aeqs2xw74m",
  "venue": "arXiv",
  "versions": []
}