{
  "abstract": "A graph on \\(n\\) vertices is called a Parter graph if there exists a nonsingular symmetric matrix, whose nonzero off-diagonal entries correspond exactly to the edges of the graph, such that all of its principal submatrices of order \\(n-1\\) are singular. Previously, a graph satisfying this condition was said to have property~(P). It was proved that, for bipartite graphs of cycle rank at most \\(3\\), being a Parter graph is equivalent to the existence of a perfect matching. We extend this result to cycle rank \\(4\\), proving that every bipartite graph of cycle rank at most \\(4\\) is a Parter graph if and only if it has a perfect matching. Furthermore, we show that this bound is sharp by constructing, for every integer \\(r\\ge5\\), a connected balanced bipartite Parter graph of cycle rank \\(r\\) that has no perfect matching.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.09869",
  "author_total": 2,
  "authors": [
    {
      "name": "G. Arunkumar"
    },
    {
      "name": "Puja Samanta"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/6vwjada847x92.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.09869",
  "fulltext_url": null,
  "id": "6vwjada847x92",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.09869",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.CO"
    ],
    [
      "openalex",
      "2608|Graph theory and applications"
    ]
  ],
  "openalex": "W7202163392",
  "pdf_url": "https://arxiv.org/pdf/2608.09869",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-13",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/6vwjada847x92.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:21:29Z",
      "native_id": "2608.09869",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.09869"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "The Cycle Rank Threshold: Perfect Matchings and Bipartite Parter Graphs",
  "title_text": "The Cycle Rank Threshold: Perfect Matchings and Bipartite Parter Graphs",
  "type": "preprint",
  "updated_at": "2026-10-30T12:21:29Z",
  "url": "https://research.draconitic.com/paper/6vwjada847x92",
  "venue": "arXiv",
  "versions": []
}