{
  "schema_version": 1,
  "problem_number": "OPG-37163",
  "title": "A Three-Round Peeling Bound for Internal Partitions of 5- and 8-Regular Graphs",
  "author": {
    "name": "Alper Ferudun",
    "affiliation": "Mercury Software GmbH",
    "url": "https://github.com/AlperTheKing"
  },
  "abstract": "Lin and Zeng proved that every 5-regular or 8-regular graph with at least 10000000 vertices has an internal partition. We refine their argument to the explicit threshold 500000. A third round of degree-p deletion in their weighted local-density argument bounds the number of degree-p vertices by N/p^3 + (2 + 2/p + 2/p^2)|X| under a local inequality on sets of at most 53 vertices. Explicit integer choices in the final small-surplus-set argument yield the improved threshold. More precisely, each side of every minimum bisection contains a ceil(r/2)-cohesive set. The existence results for these two degrees, the bisection strategy and the small-surplus-set lemma are prior work. This is a quantitative refinement, not a solution for arbitrary regular degree or an optimal-threshold claim. Exact finite computations supplement the written proof. This is an unrefereed, AI-assisted preprint; no independent review, formal verification or absolute priority certification is claimed.",
  "result_type": "COMPLETE_PROOF",
  "categories": [
    "math.CO"
  ],
  "keywords": [
    "OPG-37163",
    "internal partition",
    "friendly partition",
    "regular graph",
    "minimum bisection",
    "cohesive set"
  ],
  "manuscript_version_date": "2026-10-25",
  "publication_date": "2026-10-25",
  "publication_date_kind": "first public online release",
  "version": "1.0",
  "date_modified": "2026-10-25",
  "presentation_revision_only": false,
  "doi_archived_file_version": "1.0",
  "status": "unrefereed preprint",
  "canonical_url": "https://eulersolve.org/papers/opg-37163/",
  "pdf_url": "https://eulersolve.org/papers/opg-37163/paper.pdf?v=0910934f7c45",
  "doi": "10.5281/zenodo.23634287",
  "zenodo_record_url": "https://zenodo.org/records/23634287",
  "license": "https://creativecommons.org/licenses/by/4.0/",
  "scope_caveat": "Every finite simple undirected 5-regular or 8-regular graph with at least 500000 vertices has a nonempty ceil(r/2)-cohesive set in each side of every minimum bisection, and hence an internal partition. This improves the stated 10000000-vertex threshold of Lin and Zeng by a factor of twenty using a third deletion round. The degree-5 and degree-8 existence results are prior work; arbitrary regular degree is not resolved. Lin and Zeng already proved the degree-5 and degree-8 existence cases with threshold 10000000. The new three-round estimate improves the threshold to 500000. General regular degree remains unresolved; no new source-case closure or optimal-threshold claim is made. Absolute priority is not certified. This preprint is AI-assisted, self-audited and unrefereed; no independent or formal verification or certified novelty or absolute-priority claim is made.",
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    "source.zip": {
      "sha256": "8e79423982e3b9217d5ab17229bfd635116126983b6090c4f197d6a26c2d420d"
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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."
}