{
  "abstract": "Let $I_{n,n}$ be a fixed perfect matching of the complete bipartite graph $K_{n,n}$, and let $L_{n,1}=K_{n,n}-I_{n,n}$. A perfect set of $L_{n,1}$ is a $1$-factorization of $L_{n,1}$, that is, a set of $n-1$ pairwise edge-disjoint perfect matchings whose union is $E(L_{n,1})$; a perfect partition of $L_{n,1}$ is a partition of the set of all perfect matchings of $L_{n,1}$ into perfect sets. Brualdi, Chiang, and Li (2005) asked whether $L_{n,1}$ admits a perfect partition. Identifying both sides of the bipartition with $[n]=\\{1,2,\\ldots ,n\\}$, the perfect matchings of $L_{n,1}$ are exactly the derangements of $[n]$, the perfect sets correspond bijectively to the reduced Latin squares of order $n$, and the perfect partitions are exactly the perfect matchings of the $(n-1)$-uniform hypergraph whose vertices are the derangements of $[n]$ and whose edges are the perfect sets. We first give, for every $n$, a constructive lower bound on the maximum number of pairwise disjoint perfect sets of $L_{n,1}$. We then show that $L_{5,1}$ has exactly $42$ distinct perfect partitions and that $L_{6,1}$ has at least $82$ distinct perfect partitions; these are the first examples demonstrating that perfect partitions need not be unique. Finally, we exhibit two distinct perfect partitions of $L_{7,1}$, resolving the $n=7$ open problem raised by Brualdi et al.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2611.07118",
  "author_total": 3,
  "authors": [
    {
      "name": "Melanie J. Ferreri"
    },
    {
      "name": "Archana L. Parameswaran"
    },
    {
      "name": "Gexin Yu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/4413d4pbp78pq.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2611.07118",
  "fulltext_url": null,
  "id": "4413d4pbp78pq",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2611.07118",
  "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/2611.07118",
  "pmcid": null,
  "pmid": null,
  "published": "2026-11-04",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/4413d4pbp78pq.ris",
  "sources": [
    {
      "first_seen": "2026-11-06T18:21:34Z",
      "native_id": "2611.07118",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2611.07118"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Perfect Partitions of Regular Bipartite Graphs",
  "title_text": "Perfect Partitions of Regular Bipartite Graphs",
  "type": "preprint",
  "updated_at": "2026-11-07T18:20:21Z",
  "url": "https://research.draconitic.com/paper/4413d4pbp78pq",
  "venue": "arXiv",
  "versions": []
}