{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,12,13]],"date-time":"2025-12-13T06:54:44Z","timestamp":1765608884365,"version":"3.41.2"},"reference-count":0,"publisher":"The Electronic Journal of Combinatorics","issue":"1","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Electron. J. Combin."],"abstract":"<jats:p>A topological drawing of a graph is fan-planar if for each edge $e$ the edges crossing $e$ form a star and no endpoint of $e$ is enclosed by $e$ and its crossing edges. A fan-planar graph is a graph admitting such a drawing. Equivalently, this can be formulated by three forbidden patterns, one of which is the configuration where $e$ is crossed by two independent edges and the other two where $e$ is crossed by two incident edges in a way that encloses some endpoint of $e$. A topological drawing is simple if any two edges have at most one point in common.\r\n\u00a0Fan-planar graphs are a new member in the ever-growing list of topological graphs defined by forbidden intersection patterns, such as planar graphs and their generalizations, Tur\u00e1n-graphs and Conway's thrackle conjecture. Hence fan-planar graphs fall into an important field in combinatorial geometry with applications in various areas of discrete mathematics.\r\n\u00a0As every $1$-planar graph is fan-planar and every fan-planar graph is $3$-quasiplanar, they also fit perfectly in a recent series of works on nearly-planar graphs from the areas of graph drawing and combinatorial embeddings.\r\nIn this paper we show that every fan-planar graph on $n$ vertices has at most $5n-10$ edges, even though a fan-planar drawing may have a quadratic number of crossings. Our bound, which is tight for every $n \\geq 20$, indicates how nicely fan-planar graphs fit in the row with planar graphs ($3n-6$ edges) and $1$-planar graphs ($4n-8$ edges). With this, fan-planar graphs form an inclusion-wise largest non-trivial class of topological graphs defined by forbidden patterns, for which the maximum number of edges on $n$ vertices is known exactly.\r\nWe demonstrate that maximum fan-planar graphs carry a rich structure, which makes this class attractive for many algorithms commonly used in graph drawing. Finally, we discuss possible extensions and generalizations of these new concepts.<\/jats:p>","DOI":"10.37236\/10521","type":"journal-article","created":{"date-parts":[[2022,2,24]],"date-time":"2022-02-24T12:19:53Z","timestamp":1645705193000},"source":"Crossref","is-referenced-by-count":7,"title":["The Density of Fan-Planar Graphs"],"prefix":"10.37236","volume":"29","author":[{"given":"Michael","family":"Kaufmann","sequence":"first","affiliation":[]},{"given":"Torsten","family":"Ueckerdt","sequence":"additional","affiliation":[]}],"member":"23455","published-online":{"date-parts":[[2022,2,11]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v29i1p29\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v29i1p29\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2022,2,24]],"date-time":"2022-02-24T12:19:53Z","timestamp":1645705193000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v29i1p29"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,2,11]]},"references-count":0,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2022,1,27]]}},"URL":"https:\/\/doi.org\/10.37236\/10521","relation":{},"ISSN":["1077-8926"],"issn-type":[{"type":"electronic","value":"1077-8926"}],"subject":[],"published":{"date-parts":[[2022,2,11]]},"article-number":"P1.29"}}