{
  "schema_version": 1,
  "problem_number": "OWR-12861-021",
  "title": "Dirac Graphs Without a Spanning Near-Square of an Odd Cycle: A Negative Answer to a Question of Heinig",
  "author": {
    "name": "Alper Ferudun",
    "affiliation": "Mercury Software GmbH",
    "url": "https://github.com/AlperTheKing"
  },
  "abstract": "At the 2014 Oberwolfach workshop on combinatorics, P. Heinig asked whether, for every odd n ≥ 7, every n-vertex graph with minimum degree at least ⌈n/2⌉ contains a spanning copy of the graph obtained from the square of an n-cycle by deleting every other edge on the periphery until exactly three consecutive vertices of degree 4 remain. A positive answer would have given a structural reason why the Hamilton cycles of such graphs generate their cycle space. We show that the answer is negative for every odd n ≥ 7, under both natural readings of “periphery”, and for n = 9 under every reading. The counterexample is the complete bipartite graph K_{(n+1)/2,(n−1)/2} with a perfect matching, or a matching and one path with two edges, added inside the larger side; for n = 7 it is the graph that Heinig himself used as a positive example for the cycle-space question. The proof classifies the independent sets of size (n−1)/2 in Heinig’s graph. The cycle-space question itself has since been settled for all large odd n by Hou and Yin, and it is not affected. This is an unrefereed note.",
  "result_type": "COMPLETE_NEGATIVE_ANSWER",
  "categories": [
    "math.CO"
  ],
  "keywords": [
    "Dirac graphs",
    "spanning subgraphs",
    "square of a cycle",
    "Hamilton cycles",
    "cycle space",
    "Oberwolfach Reports",
    "counterexamples",
    "OWR-12861-021",
    "open mathematics",
    "mathematical proof",
    "math.CO"
  ],
  "manuscript_version_date": "2026-09-28",
  "publication_date": "2026-09-28",
  "publication_date_kind": "first public online release",
  "version": "1.0",
  "date_modified": "2026-09-28",
  "presentation_revision_only": false,
  "doi_archived_file_version": "1.0",
  "status": "unrefereed preprint",
  "canonical_url": "https://eulersolve.org/papers/owr-12861-021/",
  "pdf_url": "https://eulersolve.org/papers/owr-12861-021/paper.pdf?v=02ea0f37da91",
  "doi": "10.5281/zenodo.23004012",
  "zenodo_record_url": "https://zenodo.org/records/23004012",
  "license": "https://creativecommons.org/licenses/by/4.0/",
  "scope_caveat": null,
  "files": {
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    "source.zip": {
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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."
}