{
  "schema_version": 1,
  "problem_number": "KOU-21.23",
  "title": "Chordal Enhanced Power Graphs of Finite Simple Groups and Induced 4-Cycles: On Kourovka Notebook Problem 21.23",
  "author": {
    "name": "Alper Ferudun",
    "affiliation": "Mercury Software GmbH",
    "url": "https://github.com/AlperTheKing"
  },
  "abstract": "For a finite group G, the enhanced power graph E(G) has vertex set G, two distinct elements being adjacent when they generate a cyclic subgroup. Bubboloni, Fumagalli and Praeger asked (Kourovka Notebook, Problem 21.23) for which finite nonabelian simple groups E(G) has no induced cycle of a given length n ≥ 4, and for which E(G) is chordal. By their results, E(G) is chordal for PSL2(q) (q ≥ 4), the Suzuki groups, PSL3(4) and A7. We show that E(G) has an induced cycle of length at least 4 for M11, J1, the Tits group, PSL3(q) with q odd, PSU3(q) with q ≥ 3, and the Ree groups 2G2(q) with q ≥ 27; for the infinite families the cycles are given by direct constructions. Combined with the inclusion, proved by Bubboloni, Fumagalli and Praeger in an unrefereed preprint by a case analysis over the classification of finite simple groups, of all simple groups without induced 4-cycles in an explicit list, this gives: E(G) is chordal if and only if G is one of the groups of the first list. For n = 4 we show that E(G) has no induced 4-cycle for the Tits group and, using the published lists of maximal subgroups of Bray, Holt and Roney-Dougal, for PSL3(p) and PSU3(p) when p is a prime with p − 1, respectively p + 1, equal to 2^a or 3·2^a, while it has one for all other PSU3(q) and for 2G2(q) with q ≥ 27; under the same two assumptions this determines the set for n = 4. For n ≥ 5 the question remains open; we determine the least length of an induced cycle of length at least 4 for some small groups. Several steps are computer-assisted, and all certificates are included. This is an unrefereed note.",
  "result_type": "PARTIAL_RESULT",
  "categories": [
    "math.GR",
    "math.CO"
  ],
  "keywords": [
    "Kourovka Notebook",
    "enhanced power graph",
    "chordal graph",
    "induced cycle",
    "cograph",
    "finite simple groups",
    "PSL3(q)",
    "PSU3(q)",
    "Ree groups",
    "Tits group",
    "maximal subgroups",
    "computer-assisted proof",
    "partial answer",
    "UnsolvedMath",
    "KOU-21.23",
    "math.GR",
    "math.CO",
    "open mathematics"
  ],
  "manuscript_version_date": "2026-10-27",
  "publication_date": "2026-10-27",
  "publication_date_kind": "first public online release",
  "version": "1.0",
  "date_modified": "2026-10-27",
  "presentation_revision_only": false,
  "doi_archived_file_version": "1.0",
  "status": "unrefereed preprint",
  "canonical_url": "https://eulersolve.org/papers/kou-21-23/",
  "pdf_url": "https://eulersolve.org/papers/kou-21-23/paper.pdf?v=4011ad372163",
  "doi": "10.5281/zenodo.23693615",
  "zenodo_record_url": "https://zenodo.org/records/23693615",
  "license": "https://creativecommons.org/licenses/by/4.0/",
  "scope_caveat": "Part (b) and the case n = 4 are answered only granting the upper inclusion of Theorem B of an unrefereed preprint of Bubboloni, Fumagalli and Praeger (for n = 4 also the Bray-Holt-Roney-Dougal lists); part (a) for n ≥ 5 stays open; computer-assisted in part; unrefereed.",
  "files": {
    "paper.pdf": {
      "sha256": "4011ad37216372bbe007402ce5b5f8c6a0d32873dffdb709364bdc3a35551420"
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    "source.zip": {
      "sha256": "88dfc95edc3468fa6e3538049735d1525913986884c0cc0ae85d34d0028bd2fe"
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    "verification_report.md": {
      "sha256": "5f55d9110bdbcb10c2da4f4ef3fe61e5628000779abd25cbbeea3c1b67c84e1d"
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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."
}