{"paper":{"title":"An improved lower bound for the planar Tur\\'an number of cycles","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Yongxin Lan, Zi-Xia Song","submitted_at":"2022-09-03T02:34:37Z","abstract_excerpt":"The planar Tur\\'an number of a graph $H$, denoted by $ex_{_\\mathcal{P}}(n,H)$, is the largest number of edges in a planar graph on $n $ vertices without containing $H$ as a subgraph. In this paper, we continue to study the topic of \"extremal\" planar graphs initiated by Dowden [J. Graph Theory 83 (2016) 213--230]. We first obtain an improved lower bound for $ex_{_\\mathcal{P}}(n,C_k)$ for all $k\\ge 13$ and $n\\ge 5(k-6+\\lfloor{(k-1)}/2\\rfloor)(k-1)/2$; the construction for each $k$ and $n$ provides a simpler counterexample to a conjecture of Ghosh, Gy\\H{o}ri, Martin, Paulos and Xiao [arxiv:2004.1"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2209.01312","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/2209.01312/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"}