{
  "aliases": [
    "conjecture-of-kotzig",
    "regular-path-connectedness",
    "regular-path-connectivity",
    "regularly-path-connected-graph"
  ],
  "backlinks": null,
  "body": "\n## Statement\n\n**Kotzig's conjecture** is an unproven assertion in graph theory which states that finite graphs with certain properties do not exist. A graph is a $P_k$-graph if each pair of distinct vertices is connected by *exactly one* path of length $k$. Kotzig's conjecture asserts that for $k\\ge 3$ there are no finite $P_k$-graphs with two or more vertices. The conjecture was first formulated by Anton Kotzig in 1974. It has been verified for $k\\le 20$, but remains open in the general case (as of July 2026).\n",
  "commit": "68a8c3dc73fd55004b2796431c607c30960d1a85",
  "front": {
    "aliases": [
      "conjecture-of-kotzig",
      "regular-path-connectedness",
      "regular-path-connectivity",
      "regularly-path-connected-graph"
    ],
    "inventory": [
      "MHFDYGVG6V"
    ],
    "kind": "conjecture",
    "name": "Kotzig's conjecture",
    "same_as": {
      "wikidata": "Q128395030",
      "wikipedia": "Kotzig's conjecture"
    },
    "sources": [
      {
        "locator": "Kotzig's conjecture",
        "mention": "wikipedia://en/77390020",
        "note": "adopted",
        "sections": [
          "Statement"
        ],
        "wikipedia": "en:77390020@r1366924136"
      }
    ],
    "tag": "10FX3BAMEE"
  },
  "kind": "conjecture",
  "link": "https://math2.tamnd.com/node/10FX3BAMEE/",
  "links": null,
  "markdown": "https://math2.tamnd.com/node/10FX3BAMEE.md",
  "msc": null,
  "name": "Kotzig's conjecture",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/10F/10FX3BAMEE.md",
  "status": null,
  "tag": "10FX3BAMEE",
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  "url": "https://math2.tamnd.com/wiki/kotzigs-conjecture/",
  "writing": {
    "tag": "10FX3BAMEE",
    "kind": "conjecture",
    "level": "W0",
    "next": [
      {
        "what": "edge uses out",
        "have": 0,
        "want": 1
      }
    ]
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}