{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2025:ANQB3I3NTXDJO56DWNY27HDZAP","short_pith_number":"pith:ANQB3I3N","schema_version":"1.0","canonical_sha256":"03601da36d9dc69777c3b371af9c7903f40d83e221e18336dee2e1ab796fc34a","source":{"kind":"arxiv","id":"2507.16351","version":1},"attestation_state":"computed","paper":{"title":"Planar Tur\\'an number of disjoint union of $C_3$ and $C_5$","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Guiying Yan, Luyi Li, Ping Li, Qiang Zhou","submitted_at":"2025-07-22T08:33:38Z","abstract_excerpt":"The planar Tur\\'an number of $H$, denoted by $ex_{\\mathcal{P}}(n,H)$, is the maximum number of edges in an $n$-vertex $H$-free planar graph. The planar Tur\\'an number of $k\\geq 3$ vertex-disjoint union of cycles is the trivial value $3n-6$. Let $C_{\\ell}$ denote the cycle of length $\\ell$ and $C_{\\ell}\\cup C_t$ denote the union of disjoint cycles $C_{\\ell}$ and $C_t$. The planar Tur\\'an number $ex_{\\mathcal{P}}(n,H)$ is known if $H=C_{\\ell}\\cup C_k$, where $\\ell,k\\in \\{3,4\\}$. In this paper, we determine the value $ex_{\\mathcal{P}}(n,C_3\\cup C_5)=\\lfloor\\frac{8n-13}{3}\\rfloor$ and characterize"},"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":false,"formal_links_present":false},"canonical_record":{"source":{"id":"2507.16351","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2025-07-22T08:33:38Z","cross_cats_sorted":[],"title_canon_sha256":"d44d44b627d6084f2b506a3bb30a8678847606901463ad539b3a6191eb202b89","abstract_canon_sha256":"bea6cf8bb8bc5199b49d079646f8d3574a2ae53f82dff151cf5b3482fa3c5e25"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T11:41:07.913063Z","signature_b64":"Ira1IZc3mUr3xQwPjACx2S/aElo0OETrq5alRaIg2qLQErdf4YHr0EOMBpIoKmsjj9lTgIIMT5q/YcOQ/pt8CQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"03601da36d9dc69777c3b371af9c7903f40d83e221e18336dee2e1ab796fc34a","last_reissued_at":"2026-07-05T11:41:07.912665Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T11:41:07.912665Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Planar Tur\\'an number of disjoint union of $C_3$ and $C_5$","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Guiying Yan, Luyi Li, Ping Li, Qiang Zhou","submitted_at":"2025-07-22T08:33:38Z","abstract_excerpt":"The planar Tur\\'an number of $H$, denoted by $ex_{\\mathcal{P}}(n,H)$, is the maximum number of edges in an $n$-vertex $H$-free planar graph. The planar Tur\\'an number of $k\\geq 3$ vertex-disjoint union of cycles is the trivial value $3n-6$. Let $C_{\\ell}$ denote the cycle of length $\\ell$ and $C_{\\ell}\\cup C_t$ denote the union of disjoint cycles $C_{\\ell}$ and $C_t$. The planar Tur\\'an number $ex_{\\mathcal{P}}(n,H)$ is known if $H=C_{\\ell}\\cup C_k$, where $\\ell,k\\in \\{3,4\\}$. In this paper, we determine the value $ex_{\\mathcal{P}}(n,C_3\\cup C_5)=\\lfloor\\frac{8n-13}{3}\\rfloor$ and characterize"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2507.16351","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2507.16351/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"},"aliases":[{"alias_kind":"arxiv","alias_value":"2507.16351","created_at":"2026-07-05T11:41:07.912721+00:00"},{"alias_kind":"arxiv_version","alias_value":"2507.16351v1","created_at":"2026-07-05T11:41:07.912721+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2507.16351","created_at":"2026-07-05T11:41:07.912721+00:00"},{"alias_kind":"pith_short_12","alias_value":"ANQB3I3NTXDJ","created_at":"2026-07-05T11:41:07.912721+00:00"},{"alias_kind":"pith_short_16","alias_value":"ANQB3I3NTXDJO56D","created_at":"2026-07-05T11:41:07.912721+00:00"},{"alias_kind":"pith_short_8","alias_value":"ANQB3I3N","created_at":"2026-07-05T11:41:07.912721+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/ANQB3I3NTXDJO56DWNY27HDZAP","json":"https://pith.science/pith/ANQB3I3NTXDJO56DWNY27HDZAP.json","graph_json":"https://pith.science/api/pith-number/ANQB3I3NTXDJO56DWNY27HDZAP/graph.json","events_json":"https://pith.science/api/pith-number/ANQB3I3NTXDJO56DWNY27HDZAP/events.json","paper":"https://pith.science/paper/ANQB3I3N"},"agent_actions":{"view_html":"https://pith.science/pith/ANQB3I3NTXDJO56DWNY27HDZAP","download_json":"https://pith.science/pith/ANQB3I3NTXDJO56DWNY27HDZAP.json","view_paper":"https://pith.science/paper/ANQB3I3N","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2507.16351&json=true","fetch_graph":"https://pith.science/api/pith-number/ANQB3I3NTXDJO56DWNY27HDZAP/graph.json","fetch_events":"https://pith.science/api/pith-number/ANQB3I3NTXDJO56DWNY27HDZAP/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/ANQB3I3NTXDJO56DWNY27HDZAP/action/timestamp_anchor","attest_storage":"https://pith.science/pith/ANQB3I3NTXDJO56DWNY27HDZAP/action/storage_attestation","attest_author":"https://pith.science/pith/ANQB3I3NTXDJO56DWNY27HDZAP/action/author_attestation","sign_citation":"https://pith.science/pith/ANQB3I3NTXDJO56DWNY27HDZAP/action/citation_signature","submit_replication":"https://pith.science/pith/ANQB3I3NTXDJO56DWNY27HDZAP/action/replication_record"}},"created_at":"2026-07-05T11:41:07.912721+00:00","updated_at":"2026-07-05T11:41:07.912721+00:00"}