{
  "abstract": "Chromatic villainy measures the largest number of vertices that must change color to repair a rearrangement of an optimal coloring while preserving every color multiplicity. Weak villainy allows different multiplicities in the repair. On the odd cycle $C_{2r+1}$, we determine the largest repair distance from arbitrary color assignments with a prescribed color-count vector (profile) $\\mathbf m$ to proper colorings with any realizable target profile $\\mathbf p$: it equals $2r+1-\\sum_X\\min\\{p_X,\\lceil m_X/2\\rceil\\}$. In particular, every feasible profile $(a,b,c)$ has ordinary and weak repair radius $\\lfloor a/2\\rfloor+\\lfloor b/2\\rfloor+\\lfloor c/2\\rfloor$, and $B(C_{2r+1})=r$ for $r\\ge2$, answering a question of Clark et al. For arbitrary complete multipartite graphs, we give an arithmetic formula for ordinary villainy with at most $k^2$ candidates and an exact finite formula for weak villainy, where $k$ is the number of parts. Both values are fixed-parameter tractable in $k$ for binary-encoded part sizes. We also determine the maximum of each parameter among all $n$-vertex $k$-chromatic graphs. For $k\\ge3$, the ordinary maximum is $\\min\\{n,4(n-k)\\}$, except when $n=4(n-k)-1$ or $(n,k)\\in\\{(5,3),(7,3)\\}$, when it is $n-1$. For $k\\ge2$, the weak maximum is $n-\\max\\{\\lceil n/k\\rceil,2k-n\\}-\\varepsilon$, where $\\varepsilon\\in\\{0,1\\}$ is determined explicitly by $n$ and $k$. Both extrema are attained by complete multipartite graphs, and a weak extremizer needs at most three distinct part sizes.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.13580",
  "author_total": 5,
  "authors": [
    {
      "name": "Jiang Jiang"
    },
    {
      "name": "Yuhan Jiang"
    },
    {
      "name": "Yiyang Li"
    },
    {
      "name": "Lujia Wang"
    },
    {
      "name": "Aitao Zhang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/40553xqk6bztz.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.13580",
  "fulltext_url": null,
  "id": "40553xqk6bztz",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.13580",
  "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": "W7221505185",
  "pdf_url": "https://arxiv.org/pdf/2610.13580",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-09",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/40553xqk6bztz.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:28:40Z",
      "native_id": "2610.13580",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.13580"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Exact and extremal values of chromatic villainy",
  "title_text": "Exact and extremal values of chromatic villainy",
  "type": "preprint",
  "updated_at": "2026-10-30T12:28:40Z",
  "url": "https://research.draconitic.com/paper/40553xqk6bztz",
  "venue": "arXiv",
  "versions": []
}