{
  "aliases": null,
  "backlinks": [
    {
      "field": "uses",
      "target": "2REN148QWD"
    },
    {
      "field": "defines",
      "target": "KHZ89X6Y8R"
    }
  ],
  "body": "\n## Summary\n\nA word on a graph $G$ is any map $w:V(G)\\to\\mathcal C$ into a set of colors, proper or not. Its profile is the vector $\\mathbf m(w)$ whose entry $m_X(w)=|w^{-1}(X)|$ counts the vertices that get color $X$.\n\nTake $|\\mathcal C|=\\chi(G)$, the [[chromatic number|0VHJRXNAAS]] of $G$. Then each proper [[colouring|0NCPXH194Y]] with palette $\\mathcal C$ uses every color, and a profile is feasible if some such coloring has it.\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": [
      "86VNXJWJ14"
    ],
    "kind": "concept",
    "msc": [
      "05-XX"
    ],
    "name": "Words and profiles",
    "sources": [
      {
        "arxiv": "2610.13556v1",
        "licence": "cc-by",
        "locator": "Definition 2.1",
        "mention": "ax://paper/2610.13556#TOY4",
        "note": "the paper that states the item (corpus arxiv AX9.2)"
      }
    ],
    "specializes": [
      "0EPAT51DXQ"
    ],
    "tag": "28M8XD8EA9",
    "uses": [
      "0NCPXH194Y",
      "0VHJRXNAAS"
    ]
  },
  "kind": "concept",
  "link": "https://tamnd.github.io/math2/node/28M8XD8EA9/",
  "links": [
    {
      "field": "specializes",
      "target": "0EPAT51DXQ"
    },
    {
      "field": "uses",
      "target": "0NCPXH194Y"
    },
    {
      "field": "uses",
      "target": "0VHJRXNAAS"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/28M8XD8EA9.md",
  "msc": [
    "05-XX"
  ],
  "name": "Words and profiles",
  "prov": {
    "log": "https://github.com/tamnd/math2/blob/main/prov/28M/28M8XD8EA9.jsonl",
    "quarantine": null,
    "unchecked": null
  },
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/28M/28M8XD8EA9.md",
  "status": null,
  "tag": "28M8XD8EA9",
  "uri": "m2://28M8XD8EA9",
  "url": "https://tamnd.github.io/math2/wiki/words-and-profiles/",
  "writing": {
    "tag": "28M8XD8EA9",
    "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
      }
    ]
  }
}