{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Statement\n\nLet $G$ be a [[connected|0FFAYNMAJZ]] [[graph|0JD5C82QMB]].\nThen $G$ has a [[spanning tree|0FQ5B8S1ZC]].\n\n## Remarks\n\nFor a finite graph, a proof deletes edges that lie on cycles, one at a time. Such an edge is not a [[bridge|0D81KWH9ZM]], so the graph stays connected, and the process ends with a tree.\nFor an infinite graph, the proof uses Zorn's lemma, or a well-ordering of the vertices.\nThe converse also holds: a graph with a spanning tree is connected.\n\n## Explanations\n\n### L1 human\n\nIf a connected graph has a loop, you can remove one edge of the loop and the graph stays in one piece.\nRepeat until there are no loops. What is left is a spanning tree.\n",
  "commit": "6ac761d40018146541484a9e8b9eafd6ecc303dc",
  "front": {
    "kind": "statement",
    "msc": [
      "05C05"
    ],
    "name": "Every connected graph has a spanning tree",
    "role": "theorem",
    "sources": [
      {
        "locator": "Definitions",
        "url": "https://en.wikipedia.org/wiki/Spanning_tree"
      }
    ],
    "tag": "04XPQT2XTC",
    "uses": [
      "0FFAYNMAJZ",
      "0FQ5B8S1ZC"
    ]
  },
  "kind": "statement",
  "link": "https://math2.tamnd.com/node/04XPQT2XTC/",
  "links": [
    {
      "field": "uses",
      "target": "0FFAYNMAJZ"
    },
    {
      "field": "uses",
      "target": "0FQ5B8S1ZC"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/04XPQT2XTC.md",
  "msc": [
    "05C05"
  ],
  "name": "Every connected graph has a spanning tree",
  "prov": null,
  "role": "theorem",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/04X/04XPQT2XTC.md",
  "status": null,
  "tag": "04XPQT2XTC",
  "uri": "m2://04XPQT2XTC",
  "url": "https://math2.tamnd.com/wiki/every-connected-graph-has-a-spanning-tree/",
  "writing": {
    "tag": "04XPQT2XTC",
    "kind": "statement",
    "level": "W0",
    "next": [
      {
        "what": "an inventory row",
        "note": "the row of the inventory that the node accounts for",
        "have": 0,
        "want": 1
      }
    ]
  }
}