On Fan-Crossing Graphs

TL;DR

This paper explores the relationships between fan-crossing, fan-planar, and adjacency-crossing graphs, showing that adjacency-crossing graphs are a subset of fan-crossing graphs and analyzing their structural properties and density limits.

Contribution

It establishes that all adjacency-crossing graphs are fan-crossing, clarifies the differences between fan-crossing and fan-planar graphs, and compares their maximum densities.

Findings

Every adjacency-crossing graph is fan-crossing.

Fan-crossing graphs can be non-fan-planar.

Fan-crossing and fan-planar graphs have similar density bounds, at most 5n - 10 edges.

Abstract

A fan is a set of edges with a single common endpoint. A graph is fan-crossing if it admits a drawing in the plane so that each edge is crossed by edges of a fan. It is fan-planar if, in addition, the common endpoint is on the same side of the crossed edge. A graph is adjacency-crossing if it admits a drawing so that crossing edges are adjacent. Then it excludes independent crossings which are crossings by edges with no common endpoint. Adjacency-crossing allows triangle-crossings in which an edge crosses the edges of a triangle, which is excluded at fan-crossing graphs. We show that every adjacency-crossing graph is fan-crossing. Thus triangle-crossings can be avoided. On the other hand, there are fan-crossing graphs that are not fan-planar, whereas for every fan-crossing graph there is a fan-planar graph on the same set of vertices and with the same number of edges. Hence, fan-crossing and fan-planar graphs are different, but they do not differ in their density with at most 5n - 10 edges for graphs of size n.