{
  "abstract": "We prove that every triangle-divisible balanced tripartite graph whose vertex classes have size $n$ and whose partite minimum degree is at least $4n/5$ admits a fractional triangle decomposition. Combined with the multipartite decomposition theorem of Barber, Kühn, Lo, Osthus, and Taylor, this implies that, for every fixed $\\varepsilon<1/5$ and all sufficiently large $n$, every $\\varepsilon$-dense partial Latin square of order $n$ is completable, improving the previous asymptotic bound of $2/25$. The fractional decomposition theorem is finite and exact. Its proof uses a minimum-weight perfect matching to normalize an arbitrary Farkas dual weighting, followed by an explicit direct-and-two-step routing scheme whose aggregate congestion on every off-matching edge is at most one.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.31758",
  "author_total": 2,
  "authors": [
    {
      "name": "Hengzhi He"
    },
    {
      "name": "Guang Cheng"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/d8ktsmk23f89b.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.31758",
  "fulltext_url": null,
  "id": "d8ktsmk23f89b",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.31758",
  "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/2609.31758",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/d8ktsmk23f89b.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:25:17Z",
      "native_id": "2609.31758",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.31758"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Fractional Triangle Decompositions at Partite Minimum Degree $4n/5$",
  "title_text": "Fractional Triangle Decompositions at Partite Minimum Degree 4n/5",
  "type": "preprint",
  "updated_at": "2026-10-30T12:25:17Z",
  "url": "https://research.draconitic.com/paper/d8ktsmk23f89b",
  "venue": "arXiv",
  "versions": []
}