{
  "aliases": [
    "uniquely-edge-colorable-graph",
    "uniquely-total-colorable-graph"
  ],
  "backlinks": null,
  "body": "\n## Summary\n\nIn graph theory, a **uniquely colorable graph** is a $k$-chromatic [[graph|00C0BR7D7H]] that has only one possible (proper) $k$-[[coloring|0NCPXH194Y]] up to [[permutation|05RWFKVTJ7]] of the colors. Equivalently, there is only one way to partition its vertices into $k$ [[independent sets|08ANQHHG3P]] and there is no way to partition them into $k - 1$ independent sets.\n",
  "commit": "68a8c3dc73fd55004b2796431c607c30960d1a85",
  "front": {
    "aliases": [
      "uniquely-edge-colorable-graph",
      "uniquely-total-colorable-graph"
    ],
    "inventory": [
      "8H61EQJH7V"
    ],
    "kind": "concept",
    "name": "Uniquely colorable graph",
    "same_as": {
      "wikidata": "Q7886960",
      "wikipedia": "Uniquely colorable graph"
    },
    "sources": [
      {
        "locator": "Uniquely colorable graph",
        "mention": "wikipedia://en/676304",
        "note": "adopted",
        "sections": [
          "Summary"
        ],
        "wikipedia": "en:676304@r1352721310"
      }
    ],
    "tag": "HKJCK0GJHW"
  },
  "kind": "concept",
  "link": "https://math2.tamnd.com/node/HKJCK0GJHW/",
  "links": null,
  "markdown": "https://math2.tamnd.com/node/HKJCK0GJHW.md",
  "msc": null,
  "name": "Uniquely colorable graph",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/HKJ/HKJCK0GJHW.md",
  "status": null,
  "tag": "HKJCK0GJHW",
  "uri": "m2://HKJCK0GJHW",
  "url": "https://math2.tamnd.com/wiki/uniquely-colorable-graph/",
  "writing": {
    "tag": "HKJCK0GJHW",
    "kind": "concept",
    "level": "W1",
    "next": [
      {
        "what": "edge defines in",
        "note": "at least one definition",
        "have": 0,
        "want": 1
      },
      {
        "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
      }
    ]
  }
}