{"paper":{"title":"Planar Tur\\'{a}n numbers of three configurations","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Ping Li, Xuqing Bai, Zhipeng Gao","submitted_at":"2025-09-07T16:38:46Z","abstract_excerpt":"The planar Tu\\'{a}n number of $H$, denoted by $ex_{\\mathcal{P}}(n,H)$, is defined as the maximum number of edges in an $n$-vertex $H$-free planar graph. The exact value of $ex_{\\mathcal{P}}(n,H)$ remains a mystery when $H$ is large (for example, $H$ is a long path or a long cycle), while tight bounds have been established for many small planar graphs such as cycles, paths, $\\Theta$-graphs and other small graphs formed by a union of them.\n  One representative graph among such union graphs is $K_1+L$ where $L$ is a linear forest without isolated vertices.\n  Previous works solved the cases when $"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2509.06131","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/2509.06131/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"}