{
  "abstract": "For a fixed nonempty finite set $L$ of positive integers and $k=\\max L\\ge2$, let $\\binom{[n]}{L}=\\bigcup_{\\ell\\in L}\\binom{[n]}{\\ell}$ denote the non-uniform hypergraph whose edge set consists of all $\\ell$-subsets of $[n]$ for $\\ell\\in L$. A $1$-factor of a hypergraph $H$ is a collection of pairwise disjoint edges whose union is $V(H)$. We say $H$ is $1$-factorable if $E(H)$ can be partitioned into $1$-factors. In this paper, we study the 1-factorability of $\\binom{[n]}{L}$. Earlier work of He, Huang, and Ma shows that, for all sufficiently large $n$, $1$-factorability can occur only in the residue classes $0$ and $-1$ modulo $k$: it always occurs in class $0$, whereas it occurs in class $-1$ only if $k-1\\in L$. We determine exactly when $1$-factorability occurs in this exceptional class. More precisely, we define a finite gcd-peeling algorithm on $L\\setminus\\{k-1,k\\}$ and prove that $\\binom{[qk-1]}{L}$ is $1$-factorable if and only if the algorithm accepts. As a consequence, our result provides a criterion for the $1$-factorability of $\\binom{[n]}{L}$ and confirms the conjecture posed by He, Huang, and Ma.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.24695",
  "author_total": 2,
  "authors": [
    {
      "name": "Guorong Gao"
    },
    {
      "name": "Mingze Li"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/0e13q0kkdb8gp.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.24695",
  "fulltext_url": null,
  "id": "0e13q0kkdb8gp",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.24695",
  "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",
      "2607|Limits and Structures in Graph Theory"
    ]
  ],
  "openalex": "W7223006881",
  "pdf_url": "https://arxiv.org/pdf/2610.24695",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-17",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/0e13q0kkdb8gp.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:30:57Z",
      "native_id": "2610.24695",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.24695"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "On Non-Uniform Baranyai Factorizations",
  "title_text": "On Non-Uniform Baranyai Factorizations",
  "type": "preprint",
  "updated_at": "2026-10-30T12:33:44Z",
  "url": "https://research.draconitic.com/paper/0e13q0kkdb8gp",
  "venue": "arXiv",
  "versions": []
}