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. Fan-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, Turán-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. As 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. In 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. We 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.
@article{10_37236_10521,
author = {Michael Kaufmann and Torsten Ueckerdt},
title = {The density of fan-planar graphs},
journal = {The electronic journal of combinatorics},
year = {2022},
volume = {29},
number = {1},
doi = {10.37236/10521},
zbl = {1486.05166},
url = {http://geodesic.mathdoc.fr/articles/10.37236/10521/}
}
TY - JOUR
AU - Michael Kaufmann
AU - Torsten Ueckerdt
TI - The density of fan-planar graphs
JO - The electronic journal of combinatorics
PY - 2022
VL - 29
IS - 1
UR - http://geodesic.mathdoc.fr/articles/10.37236/10521/
DO - 10.37236/10521
ID - 10_37236_10521
ER -
%0 Journal Article
%A Michael Kaufmann
%A Torsten Ueckerdt
%T The density of fan-planar graphs
%J The electronic journal of combinatorics
%D 2022
%V 29
%N 1
%U http://geodesic.mathdoc.fr/articles/10.37236/10521/
%R 10.37236/10521
%F 10_37236_10521
Michael Kaufmann; Torsten Ueckerdt. The density of fan-planar graphs. The electronic journal of combinatorics, Tome 29 (2022) no. 1. doi: 10.37236/10521
