{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Statement\n\nLet $G$ be a finite simple [[graph|00C0BR7D7H]] with $m$ edges.\nLet $V$ be the vertex set of $G$.\nA *graceful labeling* of $G$ is an injective map $f \\colon V \\to \\set{0, 1, \\ldots, m}$ with the following property.\nThe edge labels $\\abs{f(u) - f(v)}$, for the edges $uv$ of $G$, are pairwise distinct.\nThen the edge labels are exactly $1, 2, \\ldots, m$.\nThe conjecture: every finite [[tree|062D9KXJB1]] has a graceful labeling.\n\n## Remarks\n\nThe conjecture is open.\nIt is also called the Ringel–Kotzig conjecture.\nIt is known for many families of trees, for example paths, stars and caterpillars, and computer searches have checked it for all trees with few vertices.\nA path $v_0 v_1 \\cdots v_m$ has the graceful labeling $0, m, 1, m - 1, 2, \\ldots$, with edge labels $m, m - 1, \\ldots, 1$.\nRosa showed that a graceful labeling of a tree $T$ with $m$ edges gives a decomposition of the complete graph $K_{2m+1}$ into $2m + 1$ copies of $T$.\nSo the conjecture would imply Ringel's conjecture on such decompositions.\nRingel's conjecture was proved for all large $m$ by Montgomery, Pokrovskiy and Sudakov, by other methods.\nNot every graph is graceful: for example the cycle $C_n$ has a graceful labeling only when $n \\equiv 0$ or $3 \\pmod 4$.\n\n## Explanations\n\n### L1 human\n\nWrite distinct numbers from $0$ to $m$ on the vertices of a tree with $m$ edges.\nLabel each edge by the difference of the numbers at its ends.\nThe labeling is graceful when the edges get the labels $1, 2, \\ldots, m$, each exactly once.\nThe conjecture says that every tree can be labeled in this way.\nIt is easy to check for small trees, but no general proof is known.\n",
  "commit": "5fc425f97f1f6b11da5892c2948ef8c5d8805665",
  "front": {
    "about": [
      "062D9KXJB1"
    ],
    "history": {
      "first_stated": "1964",
      "posed_by": "Gerhard Ringel and Anton Kotzig"
    },
    "kind": "conjecture",
    "msc": [
      "05C78",
      "05C05"
    ],
    "name": "Graceful tree conjecture",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Graceful_labeling"
      }
    ],
    "tag": "ZH3K0BH68K",
    "uses": [
      "00C0BR7D7H",
      "062D9KXJB1"
    ]
  },
  "kind": "conjecture",
  "link": "https://math2.tamnd.com/node/ZH3K0BH68K/",
  "links": [
    {
      "field": "about",
      "target": "062D9KXJB1"
    },
    {
      "field": "uses",
      "target": "00C0BR7D7H"
    },
    {
      "field": "uses",
      "target": "062D9KXJB1"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/ZH3K0BH68K.md",
  "msc": [
    "05C78",
    "05C05"
  ],
  "name": "Graceful tree conjecture",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/ZH3/ZH3K0BH68K.md",
  "status": null,
  "tag": "ZH3K0BH68K",
  "uri": "m2://ZH3K0BH68K",
  "url": "https://math2.tamnd.com/wiki/graceful-tree-conjecture/",
  "writing": {
    "tag": "ZH3K0BH68K",
    "kind": "conjecture",
    "level": "W0",
    "next": [
      {
        "what": "an inventory row",
        "note": "the row of the inventory that the node accounts for",
        "have": 0,
        "want": 1
      },
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}