{
  "abstract": "A graceful labeling of a tree with $m$ edges is a bijection from its vertices to $\\{0,1,\\ldots,m\\}$ whose edge differences are pairwise distinct. The Graceful Tree Conjecture asserts that every tree admits such a labeling; equivalently, the smallest graceful tree containing a given $m$-edge tree has $m$ edges. For this embedding problem, Sethuraman, Ragukumar and Slater proved that fewer than $4m$ edges suffice, while Sethuraman and Ragukumar observed that in practice fewer than $2m$ appear to suffice. We prove the latter bound for every tree. For any root of height $h$ with $t_h$ vertices at maximum distance, we construct a graceful tree containing the original tree with exactly $M=2m-h-t_h+1$ edges; in particular, $2m-\\operatorname{diam}(T)$ edges suffice. The construction completes a cyclic reflection of van Bussel's level labeling without increasing its span; each original edge difference $d$ is replaced by $M+1-d$. Unused vertex labels and missing edge differences are indexed by the same intermediate-level positions, allowing an explicit attachment rule that preserves the tree property. The best root for this construction is computable in linear time. Because every graceful supertree labeling restricts to a range-relaxed labeling, a uniform supertree bound of $(2-\\varepsilon)m+O(1)$ would imply such a range-relaxed bound for all trees.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2611.01340",
  "author_total": 1,
  "authors": [
    {
      "name": "Obinna Okechukwu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/3tkb2a135yzpb.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2611.01340",
  "fulltext_url": null,
  "id": "3tkb2a135yzpb",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2611.01340",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.CO"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2611.01340",
  "pmcid": null,
  "pmid": null,
  "published": "2026-11-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/3tkb2a135yzpb.ris",
  "sources": [
    {
      "first_seen": "2026-11-03T12:21:10Z",
      "native_id": "2611.01340",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2611.01340"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Graceful supertrees with at most $2m-\\mathrm{diam}(T)$ edges",
  "title_text": "Graceful supertrees with at most 2m−diam(T) edges",
  "type": "preprint",
  "updated_at": "2026-11-04T15:21:05Z",
  "url": "https://research.draconitic.com/paper/3tkb2a135yzpb",
  "venue": "arXiv",
  "versions": []
}