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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."},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":true,"formal_links_present":true},"canonical_record":{"source":{"id":"2605.17430","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2026-05-17T12:50:18Z","cross_cats_sorted":[],"title_canon_sha256":"f139dcffe9a75d13811f864728c0982e0aa14b1413dee550422e87f4b344a85a","abstract_canon_sha256":"493f6513a411d96f5250d05f8130366fc4a5fd8bb8b317482462e208ae5216ca"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-20T00:03:58.112932Z","signature_b64":"zzFVoaSX78ZBqK4DQOJRt3JH8qNUA779k01txO5nclHkwDCyxM7+AZPj+1QK27ilUT4NcPTzXVMFzLnOPvkjBQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"30ea11a8193cb8e128df130b91f63f35349cccbf49614a346e2e25c08c4687ee","last_reissued_at":"2026-05-20T00:03:58.112200Z","signature_status":"signed_v1","first_computed_at":"2026-05-20T00:03:58.112200Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"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. 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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. 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