{"paper":{"title":"Planar Tur\\'an number of quasi-double stars","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Huiqing Liu, Qin Zhao, Tian Xie","submitted_at":"2025-07-16T03:12:10Z","abstract_excerpt":"Given a graph H, we call a graph $\\textit{H-free}$ if it does not contain H as a subgraph. The planar Tur\\'an number of a graph H, denoted by $ex_{\\mathcal{P}}(n, H)$, is the maximum number of edges in a planar H-free graph on n vertices. A (h,k)-quasi-double star $W_{h,k}$, obtained from a path $P_3=v_1v_2v_3$ by adding h leaves and k leaves to the vertices $v_1$ and $v_3$, respectively, is a subclass of caterpillars. In this paper, we study $ex_{\\mathcal{P}}(n,W_{h,k})$ for all $1\\le h\\le 2\\le k\\le 5$, and obtain some tight bounds $ex_{\\mathcal{P}}(n,W_{h,k})\\leq\\frac{3(h+k)}{h+k+2}n$ for $3"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2507.11860","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/2507.11860/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"}