{
  "abstract": "A connected graph with at least two vertices is matching covered if each of its edges is contained in a perfect matching. A matching covered graph is solid if every separating cut in it is a tight cut. A matching covered graph which is free of nontrivial tight cuts is a brick if it is nonbipartite. Every bipartite matching covered graph is solid. Lucchesi and Murty conjectured that there exists a positive integer $N$ such that, for every integer $n\\ge N$, the maximum number of edges in a simple solid matching covered graph on $2n$ vertices is $n^2$. In this paper, we disprove this Conjecture, and show that the maximum number of edges of a simple solid matching covered graph on $2n$ ($n\\ge2$) vertices is $n^2+2$. Moreover, we characterize the graphs attaining this bound. In addition, we prove that every simple solid brick of order $2n$ has at most $n^2$ edges for $n\\ge4$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.07733",
  "author_total": 2,
  "authors": [
    {
      "name": "Tong Zhang"
    },
    {
      "name": "Wei Li"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/51v2s9qva08h4.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.07733",
  "fulltext_url": null,
  "id": "51v2s9qva08h4",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.07733",
  "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",
      "1703|Advanced Graph Theory Research"
    ]
  ],
  "openalex": "W7221031931",
  "pdf_url": "https://arxiv.org/pdf/2610.07733",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-06",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/51v2s9qva08h4.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:26Z",
      "native_id": "2610.07733",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.07733"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "The maximum size of simple solid matching covered graphs",
  "title_text": "The maximum size of simple solid matching covered graphs",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:26Z",
  "url": "https://research.draconitic.com/paper/51v2s9qva08h4",
  "venue": "arXiv",
  "versions": []
}