{"paper":{"title":"Extremal $H$-free planar graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Yongtang Shi, Yongxin Lan, Zi-Xia Song","submitted_at":"2018-08-04T14:41:45Z","abstract_excerpt":"Given a graph $H$, a graph is $H$-free if it does not contain $H$ as a subgraph. We continue to study the topic of \"extremal\" planar graphs, that is, how many edges can an $H$-free planar graph on $n$ vertices have? We define $ex_{_\\mathcal{P}}(n,H)$ to be the maximum number of edges in an $H$-free planar graph on $n $ vertices. We first obtain several sufficient conditions on $H$ which yield $ex_{_\\mathcal{P}}(n,H)=3n-6$ for all $n\\ge |V(H)|$. We discover that the chromatic number of $H$ does not play a role, as in the celebrated Erd\\H{o}s-Stone Theorem. We then completely determine $ex_{_\\ma"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1808.01487","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":""},"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"}