Arrangements of orthogonal circles with many intersections

TL;DR

This paper investigates arrangements of orthogonal circles, proving planarity of their intersection graphs under certain conditions and establishing bounds on the maximum number of edges and triangles in these graphs.

Contribution

It establishes planarity of intersection graphs for non-nested orthogonal circle arrangements and provides bounds on edges and triangles for general arrangements.

Findings

Intersection graphs are planar when no two circles are nested.

Maximum edges in intersection graphs are bounded between 4n - O(√n) and (4+5/11)n.

Number of triangles is at least (3+5/9)n - O(√n).

Abstract

An arrangement of circles in which circles intersect only in angles of $\pi/2$ is called an \emph{arrangement of orthogonal circles}. We show that in the case that no two circles are nested, the intersection graph of such an arrangement is planar. The same result holds for arrangement of circles that intersect in an angle of at most $\pi/2$. For the general case we prove that the maximal number of edges in an intersection graph of an arrangement of orthogonal circles lies in between $4n - O\left(\sqrt{n}\right)$ and $\left(4+\frac{5}{11}\right)n$, for $n$ being the number of circles. Based on the lower bound we can also improve the bound for the number of triangles in arrangements of orthogonal circles to $(3 + 5/9)n-O\left(\sqrt{n}\right)$.