Drawing Graphs with Circular Arcs and Right Angle Crossings.

In a RAC drawing of a graph, vertices are represented by points in the plane, adjacent vertices are connected by line segments, and crossings must form right angles. Graphs that admit such drawings are RAC graphs. RAC graphs are beyond-planar graphs and have been studied extensively. In particular, it is known that a RAC graph with n vertices has at most 4n - 10 edges. We introduce a superclass of RAC graphs, which we call arc-RAC graphs. In an arc-RAC drawing, edges are drawn as circular arcs whereas crossings must still form right angles. We provide a Turan-type result showing that an arc-RAC graph with n vertices has at most 14n - 12 edges and that there are n-vertex arc-RAC graphs with 4.5n - o(n) edges.