{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,3]],"date-time":"2026-06-03T03:32:22Z","timestamp":1780457542962,"version":"3.54.1"},"reference-count":0,"publisher":"The Electronic Journal of Combinatorics","issue":"2","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Electron. J. Combin."],"abstract":"<jats:p>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 initiated by Dowden [J. Graph Theory\u00a0 83 (2016) 213\u2013230], 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\u00a0 $ex_{_\\mathcal{P}}(n,H)=3n-6$ for all $n\\ge |V(H)|$. We discover that the chromatic number of $H$ does not play a role, as in the celebrated Erd\u0151s-Stone Theorem.\u00a0 We then completely determine $ex_{_\\mathcal{P}}(n,H)$ when $H$ is a wheel or a star. Finally, we examine the case when $H$ is a $(t, r)$-fan, that is, $H$ is isomorphic to\u00a0 $K_1+tK_{r-1}$, where $t\\ge2$ and $r\\ge 3$ are integers. However, determining $ex_{_\\mathcal{P}}(n,H)$, when $H$ is a planar subcubic graph, remains wide open.<\/jats:p>","DOI":"10.37236\/8255","type":"journal-article","created":{"date-parts":[[2020,1,10]],"date-time":"2020-01-10T07:07:52Z","timestamp":1578640072000},"source":"Crossref","is-referenced-by-count":15,"title":["Extremal $H$-Free Planar Graphs"],"prefix":"10.37236","volume":"26","author":[{"given":"Yongxin","family":"Lan","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Yongtang","family":"Shi","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-6183-0110","authenticated-orcid":false,"given":"Zi-Xia","family":"Song","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"23455","published-online":{"date-parts":[[2019,5,3]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v26i2p11\/7824","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v26i2p11\/7824","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,1,17]],"date-time":"2020-01-17T04:14:32Z","timestamp":1579234472000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v26i2p11"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2019,5,3]]},"references-count":0,"journal-issue":{"issue":"2","published-online":{"date-parts":[[2019,4,5]]}},"URL":"https:\/\/doi.org\/10.37236\/8255","relation":{},"ISSN":["1077-8926"],"issn-type":[{"value":"1077-8926","type":"electronic"}],"subject":[],"published":{"date-parts":[[2019,5,3]]},"article-number":"P2.11"}}