The Density of Fan-Planar Graphs

A topological drawing of a graph is fan-planar if for each edge e the edges crossing e have a common endpoint on the same side of e, and a fan-planar graph is a graph admitting such a drawing. Equivalently, this can be formulated by two forbidden patterns, one of which is the configuration where e is crossed by two independent edges and the other where e is crossed by incident edges with the common endpoint on different sides of e. In particular every edge of a fan-planar graph is crossed only by the edges of a star. A topological drawing is simple if any two edges have at most one point in common. The class of fan-planar graphs is a natural variant of other classes defined by forbidden intersection patterns in a topological drawing of the graph. So every 1-planar graph is also fan-planar, and every fan-planar graph is also quasiplanar, where both inclusions are strict. Fan-planar graphs also fit perfectly in a recent series of work on nearly-planar graphs from the area of graph drawing and combinatorial embeddings. For topologically defined graph classes, one of the most fundamental questions asks for the maximum number of edges in any such graph with n vertices. We prove that every n-vertex graph without loops and parallel edges that admits a simple fan-planar drawig has at most 5n− 10 edges and that this bound is tight for every n ≥ 20. Furthermore we discuss possible extensions and generalizations of these new concepts.