# Arrangements of Pseudocircles: Triangles and Drawings

@article{Felsner2017ArrangementsOP,
title={Arrangements of Pseudocircles: Triangles and Drawings},
author={Stefan Felsner and Manfred Scheucher},
journal={Discrete \& Computational Geometry},
year={2017},
volume={65},
pages={261 - 278}
}
• Published 21 August 2017
• Mathematics
• Discrete & Computational Geometry
A pseudocircle is a simple closed curve on the sphere or in the plane. The study of arrangements of pseudocircles was initiated by Grünbaum, who defined them as collections of simple closed curves that pairwise intersect in exactly two crossings. Grünbaum conjectured that the number of triangular cells $$p_3$$ p 3 in digon-free arrangements of n pairwise intersecting pseudocircles is at least $$2n-4$$ 2 n - 4 . We present examples to disprove this conjecture. With a recursive construction based…
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It is shown that there are exactly four non-circularizable arrangements of 5 pseudocircles, exactly one of them is intersecting and that every digon-free intersecting arrangement of circles contains at least $2n-4$ triangles.