{
  "aliases": null,
  "backlinks": [
    {
      "field": "defines",
      "target": "2REN148QWD"
    }
  ],
  "body": "\n## Summary\n\nA repair radius measures how far a colouring of a graph $G$ with a given profile of class sizes can be from a proper one.\n\nThe radius $R_G(m)$ is the largest Hamming [[distance from a point to a set|07TXYYVYZR]], over colourings with profile $m$, to the set of [[proper colourings|0874138YP6]] with profile $m$.\n\nThe weak radius $R^w_G(m)$ lets the nearest proper colouring have any profile.\n\nThe villainy $B(G)$ and the weak villainy $B_w(G)$ are the maxima of these radii over feasible profiles, and both are zero for the empty graph.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "history": {
      "notes": "The source is arXiv:2610.13556v1 (2026-10), by Jiang Jiang, Yuhan Jiang, Yiyang Li, Lujia Wang, Aitao Zhang. A paper also restates older results, so this is not the first statement until a person or a later pass confirms it."
    },
    "inventory": [
      "GY5XXNYENM"
    ],
    "kind": "concept",
    "msc": [
      "05-XX"
    ],
    "name": "Repair radii and villainy",
    "sources": [
      {
        "arxiv": "2610.13556v1",
        "licence": "cc-by",
        "locator": "Definition 2.2",
        "mention": "ax://paper/2610.13556#L0DK",
        "note": "the paper that states the item (corpus arxiv AX9.2)"
      }
    ],
    "tag": "EBNHD7FQ23",
    "uses": [
      "0874138YP6",
      "07TXYYVYZR"
    ]
  },
  "kind": "concept",
  "link": "https://math2.tamnd.com/node/EBNHD7FQ23/",
  "links": [
    {
      "field": "uses",
      "target": "0874138YP6"
    },
    {
      "field": "uses",
      "target": "07TXYYVYZR"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/EBNHD7FQ23.md",
  "msc": [
    "05-XX"
  ],
  "name": "Repair radii and villainy",
  "prov": {
    "log": "https://github.com/tamnd/math2/blob/main/prov/EBN/EBNHD7FQ23.jsonl",
    "quarantine": null,
    "unchecked": null
  },
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/EBN/EBNHD7FQ23.md",
  "status": null,
  "tag": "EBNHD7FQ23",
  "uri": "m2://EBNHD7FQ23",
  "url": "https://math2.tamnd.com/wiki/repair-radii-and-villainy/",
  "writing": {
    "tag": "EBNHD7FQ23",
    "kind": "concept",
    "level": "W1",
    "next": [
      {
        "what": "edge instance_of in",
        "note": "at least two examples",
        "have": 0,
        "want": 2
      },
      {
        "what": "edge not_instance_of in",
        "note": "at least one non-example",
        "have": 0,
        "want": 1
      },
      {
        "what": "edge specializes both",
        "note": "the more general notions and the special cases",
        "have": 0,
        "want": 1
      }
    ]
  }
}