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We determine the planar Tur\\'an number of $C_{3}\\text{-}C_{3}$ and $C_{3}\\text{-}C_{4}$, where $C_{k}\\text{-}C_{\\ell}$ denotes the graph consisting of two disjoint cycles $C_k$ with an edge connecting them."},"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":"2411.18487","kind":"arxiv","version":2},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2024-11-27T16:30:06Z","cross_cats_sorted":[],"title_canon_sha256":"69b4543da1cbb21052814c5ce5719e6d93ed91fb9cdbb241c2f059ab25e9bbf1","abstract_canon_sha256":"238fe3d73dad56efaa65a1b36b5df25fdd744cce20f2b7db999eeb3fa14c148e"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T11:07:17.145786Z","signature_b64":"2nl4ZCSGBfxm5nu+00JbOtf9w9/ZrXMoWJNTIPYTgZPzYTB1jOJIzbYeIM5L6pBGNPgpD3Ibmg04ydkwlCZFDQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"0a7633222d8f6c2d48d9f23c3db31ac4f126190e0130a2b697ab31ae5d730160","last_reissued_at":"2026-07-05T11:07:17.145266Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T11:07:17.145266Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Planar Tur\\'an number of two adjacent cycles","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Guiying Yan, Qiang Zhou, Xinzhe Song","submitted_at":"2024-11-27T16:30:06Z","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 $k(k\\geq 3)$ vertex-disjoint union of cycles is the trivial value $3n-6$. We determine the planar Tur\\'an number of $C_{3}\\text{-}C_{3}$ and $C_{3}\\text{-}C_{4}$, where $C_{k}\\text{-}C_{\\ell}$ denotes the graph consisting of two disjoint cycles $C_k$ with an edge connecting them."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2411.18487","kind":"arxiv","version":2},"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/2411.18487/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":"2411.18487","created_at":"2026-07-05T11:07:17.145341+00:00"},{"alias_kind":"arxiv_version","alias_value":"2411.18487v2","created_at":"2026-07-05T11:07:17.145341+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2411.18487","created_at":"2026-07-05T11:07:17.145341+00:00"},{"alias_kind":"pith_short_12","alias_value":"BJ3DGIRNR5WC","created_at":"2026-07-05T11:07:17.145341+00:00"},{"alias_kind":"pith_short_16","alias_value":"BJ3DGIRNR5WC2SGZ","created_at":"2026-07-05T11:07:17.145341+00:00"},{"alias_kind":"pith_short_8","alias_value":"BJ3DGIRN","created_at":"2026-07-05T11:07:17.145341+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/BJ3DGIRNR5WC2SGZ6I6D3MY2YT","json":"https://pith.science/pith/BJ3DGIRNR5WC2SGZ6I6D3MY2YT.json","graph_json":"https://pith.science/api/pith-number/BJ3DGIRNR5WC2SGZ6I6D3MY2YT/graph.json","events_json":"https://pith.science/api/pith-number/BJ3DGIRNR5WC2SGZ6I6D3MY2YT/events.json","paper":"https://pith.science/paper/BJ3DGIRN"},"agent_actions":{"view_html":"https://pith.science/pith/BJ3DGIRNR5WC2SGZ6I6D3MY2YT","download_json":"https://pith.science/pith/BJ3DGIRNR5WC2SGZ6I6D3MY2YT.json","view_paper":"https://pith.science/paper/BJ3DGIRN","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2411.18487&json=true","fetch_graph":"https://pith.science/api/pith-number/BJ3DGIRNR5WC2SGZ6I6D3MY2YT/graph.json","fetch_events":"https://pith.science/api/pith-number/BJ3DGIRNR5WC2SGZ6I6D3MY2YT/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/BJ3DGIRNR5WC2SGZ6I6D3MY2YT/action/timestamp_anchor","attest_storage":"https://pith.science/pith/BJ3DGIRNR5WC2SGZ6I6D3MY2YT/action/storage_attestation","attest_author":"https://pith.science/pith/BJ3DGIRNR5WC2SGZ6I6D3MY2YT/action/author_attestation","sign_citation":"https://pith.science/pith/BJ3DGIRNR5WC2SGZ6I6D3MY2YT/action/citation_signature","submit_replication":"https://pith.science/pith/BJ3DGIRNR5WC2SGZ6I6D3MY2YT/action/replication_record"}},"created_at":"2026-07-05T11:07:17.145341+00:00","updated_at":"2026-07-05T11:07:17.145341+00:00"}