{"paper":{"title":"Triangles in graphs without the expansion of $4$-cycle","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"The expansion of the 4-cycle is the only counterexample to a conjecture on the maximum number of triangles in graphs avoiding expanded paths and cycles.","cross_cats":[],"primary_cat":"math.CO","authors_text":"Jialei Song, Long-tu Yuan, Qi Wu","submitted_at":"2026-05-17T12:50:18Z","abstract_excerpt":"The expansion $F^{\\triangle}$ of a graph $F$ is the graph obtained from $F$ by replacing each edge with a triangle. Lv \\etal proposed a conjecture on the maximum number of triangles in a graph without $P_k^{\\triangle}$ or $C_k^{\\triangle}$ for every $k \\ge 4$. Their conjecture was confirmed in previous work for $P_k^{\\triangle}$ when $k \\ge 4$ and $C_k^{\\triangle}$ when $k \\ge 5$. In this note, we resolve the remaining case $C_4^{\\triangle}$, demonstrating that this is the only counterexample to their conjecture."},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"We resolve the remaining case C_4^Δ, demonstrating that this is the only counterexample to their conjecture.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The validity of the conjecture for all previously confirmed cases (P_k^Δ for k≥4 and C_k^Δ for k≥5) and the standard definition of the graph expansion F^Δ as replacing each edge by a triangle.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"Resolves the remaining C_4^Δ case of the conjecture on maximum triangles in graphs without P_k^Δ or C_k^Δ for k≥4, showing it is the only counterexample.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"The expansion of the 4-cycle is the only counterexample to a conjecture on the maximum number of triangles in graphs avoiding expanded paths and cycles.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"1715c2182cffca5ef4217f126d4263e3df98b4e8624a3d39e84c2a51e786e3e4"},"source":{"id":"2605.17430","kind":"arxiv","version":1},"verdict":{"id":"f1f8e99f-e94b-4c28-b114-d3ac71adf2db","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-19T22:52:34.479223Z","strongest_claim":"We resolve the remaining case C_4^Δ, demonstrating that this is the only counterexample to their conjecture.","one_line_summary":"Resolves the remaining C_4^Δ case of the conjecture on maximum triangles in graphs without P_k^Δ or C_k^Δ for k≥4, showing it is the only counterexample.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The validity of the conjecture for all previously confirmed cases (P_k^Δ for k≥4 and C_k^Δ for k≥5) and the standard definition of the graph expansion F^Δ as replacing each edge by a triangle.","pith_extraction_headline":"The expansion of the 4-cycle is the only counterexample to a conjecture on the maximum number of triangles in graphs avoiding expanded paths and cycles."},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2605.17430/integrity.json","findings":[],"available":true,"detectors_run":[{"name":"doi_compliance","ran_at":"2026-05-19T23:01:27.696573Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"doi_title_agreement","ran_at":"2026-05-19T23:01:19.605190Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"claim_evidence","ran_at":"2026-05-19T21:41:57.730763Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"ai_meta_artifact","ran_at":"2026-05-19T21:33:23.678092Z","status":"skipped","version":"1.0.0","findings_count":0}],"snapshot_sha256":"1e551f857514a8aa8741400c18ddfea8d55ab65e3b492e481c5eae2dc24c6256"},"references":{"count":11,"sample":[{"doi":"","year":2016,"title":"N. Alon and C. Shikhelman. ManyTcopies inH-free graphs.J. Combin. Theory Ser. B, 121:146–172, 2016. 2","work_id":"edf56659-8c96-4675-85b1-4979e5d89d4b","ref_index":1,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":1962,"title":"P. Erd˝ os. On the number of complete subgraphs contained in certain graphs.Magyar Tud. Akad. Mat. Kutat´ o Int. K¨ ozl., 7:459–464, 1962. 2","work_id":"4ea16ec3-92c4-46e5-b7a4-f4d7172a90cb","ref_index":2,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2024,"title":"D. Gerbner and B. Patk´ os. Generalized Tur´ an results for intersecting cliques.Discrete Math., 347(1):113710, 2024. 2","work_id":"1e5835fe-e08e-4fdd-b7ce-8c9568472fbe","ref_index":3,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2015,"title":"A. Kostochka, D. Mubayi, and J. Verstra¨ ete. Tur´ an problems and shadows I: Paths and cycles.J. Combin. Theory Ser. A, 129:57–79, 2015. 2, 3","work_id":"7af2a8f8-2fa1-4df0-b802-a239ab7adcec","ref_index":4,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2017,"title":"A. Kostochka, D. Mubayi, and J. Verstra¨ ete. Tur´ an problems and shadows II: Trees. J. Combin. Theory Ser. B, 122:457–478, 2017. 2 8","work_id":"0a8ac9fe-9f9a-42d6-bba8-5f9b97901df0","ref_index":5,"cited_arxiv_id":"","is_internal_anchor":false}],"resolved_work":11,"snapshot_sha256":"5f61a45918d452bf4ede5fdcbdb1cd9e2a735a53a38a4126f73448604448e3f4","internal_anchors":0},"formal_canon":{"evidence_count":2,"snapshot_sha256":"46a4e99f4cfc5ed11fecd0d15aa3b1affd36e4f82b5cb89aff6221bb06b967ef"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}