{
  "abstract": "For a fixed $r$, let $f_r(n)$ denote the minimum number of complete $r$-partite $r$-uniform hypergraphs whose edge sets partition the complete $r$-uniform hypergraph on $n$ vertices. The Graham-Pollak theorem asserts that $f_2(n)=n-1$. Let $c_r$ be the smallest constant such that $f_r(n)\\le c_r(1+o(1))\\binom{n}{\\lfloor r/2\\rfloor}$. We give two constructions that improve upper bounds for $f_r(n)$. The first partitions $E(K_4)\\times E(K_{17})$ into $44$ products of edge sets of complete bipartite graphs and yields $c_4\\le11/12$, improving the previous bound of $c_4\\le14/15$. The second gives an asymptotically improved exact cover for odd $r$. Combining these constructions, we prove that $c_r<1$ for every odd integer $r\\ge65$, which improves upon the previous computer-assisted result of $113$. We further obtain the improved asymptotic bound $c_r\\le \\frac{r}{15}\\left(\\frac{11}{12}\\right)^{r/4}+o(1)$ for general $r$. These results narrow the gap towards resolving the major open problem of determining whether $c_5<1$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2607.15658",
  "author_total": 1,
  "authors": [
    {
      "name": "Anand Babu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/fhzxzfwxwxhej.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2607.15658",
  "fulltext_url": null,
  "id": "fhzxzfwxwxhej",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2607.15658",
  "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",
      "cs.DM"
    ],
    [
      "arxiv",
      "math.CO"
    ],
    [
      "openalex",
      "2607|Limits and Structures in Graph Theory"
    ]
  ],
  "openalex": "W7169791392",
  "pdf_url": "https://arxiv.org/pdf/2607.15658",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-14",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/fhzxzfwxwxhej.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:21:29Z",
      "native_id": "2607.15658",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2607.15658"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "New bounds on the Graham-Pollak theorem for hypergraphs",
  "title_text": "New bounds on the Graham-Pollak theorem for hypergraphs",
  "type": "preprint",
  "updated_at": "2026-10-30T12:21:29Z",
  "url": "https://research.draconitic.com/paper/fhzxzfwxwxhej",
  "venue": "arXiv",
  "versions": []
}