{"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"}