{
  "abstract": "Let \\(\\cthreshold(n)\\) be the largest integer \\(q\\) such that every loopless digraph on \\(n\\) vertices with at most \\(q\\) arcs is isomorphic to a spanning subdigraph of a self-complementary digraph of order \\(n\\). We prove that \\(\\cthreshold(6)=7\\). The upper bound is witnessed by \\[ \\bK{3}\\dunion (x\\longrightarrow y\\longrightarrow z), \\] and follows from a direct argument with a self-complementing permutation. We also determine the complete eight-arc obstruction layer: it consists of five isomorphism classes, or three after converse digraphs are identified. All five are arc-minimal. Each nevertheless packs with an isomorphic copy of itself, so ordinary packing is strictly weaker than same-order self-complementary completion already at this first failure layer.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.20231",
  "author_total": 5,
  "authors": [
    {
      "name": "Xinan Dai"
    },
    {
      "name": "Wenhao Deng"
    },
    {
      "name": "Yingdong Shi"
    },
    {
      "name": "Tailin Wu"
    },
    {
      "name": "Yuchen Yang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/ekg8a6jxhz8wb.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.20231",
  "fulltext_url": null,
  "id": "ekg8a6jxhz8wb",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.20231",
  "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",
      "cs.AI"
    ],
    [
      "arxiv",
      "math.CO"
    ],
    [
      "openalex",
      "1703|Advanced Graph Theory Research"
    ]
  ],
  "openalex": "W7213604215",
  "pdf_url": "https://arxiv.org/pdf/2609.20231",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/ekg8a6jxhz8wb.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:22:23Z",
      "native_id": "2609.20231",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.20231"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Self-complementary completions on six vertices",
  "title_text": "Self-complementary completions on six vertices",
  "type": "preprint",
  "updated_at": "2026-10-30T12:22:23Z",
  "url": "https://research.draconitic.com/paper/ekg8a6jxhz8wb",
  "venue": "arXiv",
  "versions": []
}