Special issue on Selected papers from the Twenty-nineth International Symposium on Graph Drawing and Network Visualization, GD 2021
Arrangements of orthogonal circles with many intersections
Sarah Carmesin and André Schulz
Vol. 27, no. 2, pp. 49-70, 2023. Regular paper.
Abstract An arrangement of circles in which circles intersect only in angles of $\pi/2$ is called an 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 case where circles can be nested 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 lower bound for the number of triangles in arrangements of orthogonal circles to $(3 + \frac{5}{9})n-O\left(\sqrt{n}\right)$.

 This work is licensed under the terms of the CC-BY license.
Submitted: November 2021.
Reviewed: March 2022.
Revised: March 2022.
Reviewed: October 2022.
Revised: November 2022.
Accepted: November 2022.
Final: November 2022.
Published: February 2023.
Communicated by Ignaz Rutter and Helene Purchase
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