{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Statement\n\nIn graph theory, the **Nash-Williams theorem** is a **tree-packing** theorem that describes how many edge-disjoint [[spanning trees|0FQ5B8S1ZC]] (and more generally forests) a graph can have:\n\n\u003e A graph *G* has *t* edge-disjoint spanning trees iff for every partition $V_1, \\ldots, V_k \\subset V(G)$ where $V_i \\neq \\emptyset$ there are at least *t*(*k* − 1) crossing edges.\n\nThe theorem was proved independently by Tutte and Nash-Williams, both in 1961. In 2012, Kaiser gave a short elementary proof.\n",
  "commit": "68a8c3dc73fd55004b2796431c607c30960d1a85",
  "front": {
    "inventory": [
      "PR9BT1BA40"
    ],
    "kind": "statement",
    "name": "Nash-Williams theorem",
    "role": "proposition",
    "same_as": {
      "wikidata": "Q65066994",
      "wikipedia": "Nash-Williams theorem"
    },
    "sources": [
      {
        "locator": "Nash-Williams theorem",
        "mention": "wikipedia://en/60654680",
        "note": "adopted",
        "sections": [
          "Statement"
        ],
        "wikipedia": "en:60654680@r1332584259"
      }
    ],
    "tag": "BYJ5EZMP6W"
  },
  "kind": "statement",
  "link": "https://math2.tamnd.com/node/BYJ5EZMP6W/",
  "links": null,
  "markdown": "https://math2.tamnd.com/node/BYJ5EZMP6W.md",
  "msc": null,
  "name": "Nash-Williams theorem",
  "prov": null,
  "role": "proposition",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/BYJ/BYJ5EZMP6W.md",
  "status": null,
  "tag": "BYJ5EZMP6W",
  "uri": "m2://BYJ5EZMP6W",
  "url": "https://math2.tamnd.com/wiki/nash-williams-theorem/",
  "writing": {
    "tag": "BYJ5EZMP6W",
    "kind": "statement",
    "level": "W0",
    "next": [
      {
        "what": "edge uses out",
        "note": "each definition that the statement needs",
        "have": 0,
        "want": 1
      }
    ]
  }
}