{
  "schema_version": 1,
  "problem_number": "AIM-COMBINATORICS-0183",
  "title": "A Sharp Three-Fifths Bound for Seymour Vertices in Three-Regular Oriented Graphs",
  "author": {
    "name": "Alper Ferudun",
    "affiliation": "Mercury Software GmbH",
    "url": "https://github.com/AlperTheKing"
  },
  "abstract": "Let D be a finite simple oriented graph with indegree and outdegree three at every vertex. A Seymour vertex has at least as many exact second out-neighbors as first out-neighbors. We prove that at least ceil(3|V(D)|/5) vertices are Seymour vertices. If B is the set of the remaining vertices, the sum of |N^{++}(v)|-3 over all vertices is at least ceil(|B|/2). Both constants are sharp, even for strongly connected graphs.\n\nWe classify all graphs attaining the exact, unrounded three-fifths proportion. They are expansions of loopless two-in/two-out directed multigraphs: replace each vertex with a directed triangle and each arc with a new vertex joined completely to its two endpoint triangles. An exact surplus formula then characterizes simultaneous equality in both bounds. The proofs use local arc capacities, indegree saturation and equality in an incidence count. The package includes the six-page English manuscript, standalone LaTeX and reproducible finite checks.\n\nThis is a complete theorem for the three-regular class associated with AIM-COMBINATORICS-0183 in UnsolvedMath v1.6.0. It does not resolve the mean second-neighborhood inequality for arbitrary Eulerian oriented graphs, including mixed degrees at most three. The source record remains partial. The manuscript is AI-assisted, self-audited and unrefereed; independent review, formal verification and absolute priority are not claimed.",
  "result_type": "COMPLETE_PROOF",
  "categories": [
    "math.CO"
  ],
  "keywords": [
    "oriented graph",
    "Eulerian digraph",
    "second neighborhood",
    "Seymour vertex",
    "regular digraph",
    "extremal proportion",
    "equality case"
  ],
  "manuscript_version_date": "2026-10-10",
  "publication_date": "2026-10-10",
  "publication_date_kind": "first public online release",
  "version": "1.0",
  "date_modified": "2026-10-10",
  "presentation_revision_only": false,
  "doi_archived_file_version": "1.0",
  "status": "unrefereed preprint",
  "canonical_url": "https://eulersolve.org/papers/aim-combinatorics-0183/",
  "pdf_url": "https://eulersolve.org/papers/aim-combinatorics-0183/paper.pdf?v=f1033c0bb3f4",
  "doi": "10.5281/zenodo.23278659",
  "zenodo_record_url": "https://zenodo.org/records/23278659",
  "license": "https://creativecommons.org/licenses/by/4.0/",
  "scope_caveat": "For every finite simple oriented three-in/three-out regular graph, at least ceil(3n/5) vertices have at least three exact second out-neighbors, and the total second-neighborhood surplus is at least ceil(|B|/2). Both coefficients are sharp. Equality 5|B|=2n is classified by loopless two-in/two-out directed multigraph triangle expansions; simultaneous equality is precisely the alternating triangle/two-vertex cycle family. The general Eulerian mean problem remains unresolved. This preprint is self-audited and unrefereed, without certified priority or formal verification.",
  "files": {
    "paper.pdf": {
      "sha256": "f1033c0bb3f4a3bab9c038b3a312b5164b952d5cb10365628134e3c4dcf72e8e"
    },
    "source.zip": {
      "sha256": "708b4bd3689da39e602b04bd946ad1987daa4b9b58a487ec6692cc331ec336ec"
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    "verification_report.md": {
      "sha256": "9b0720b8df86d862db43bb0455334d82cccfc1268d88cddafcbba753e1c39406"
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  },
  "ai_use_disclosure": "AI-assisted tools supported research, computation, proof development, and manuscript preparation. The author remains responsible for all claims and the final text."
}