Abstract

A pseudocircle is a simple closed curve on the sphere or in the plane. The study of arrangements of pseudocircles was initiated by Grunbaum, who defined them as collections of simple closed curves that pairwise intersect in exactly two crossings. Grunbaum conjectured that the number of triangular cells \(p_3\) in digon-free arrangements of n pairwise intersecting pseudocircles is at least \(2n-4\). We present examples to disprove this conjecture. With a recursive construction based on an example with 12 pseudocircles and 16 triangles we obtain a family with \(p_3(\mathscr {A})/n \rightarrow 16/11 = 1.\overline{45}\). We expect that the lower bound \(p_3(\mathscr {A}) \ge 4n/3\) is tight for infinitely many simple arrangements. It may however be that digon-free arrangements of n pairwise intersecting circles indeed have at least \(2n-4\) triangles.

Highlights

  • Arrangements of pseudocircles generalize arrangements of circles in the same vein as arrangements of pseudolines generalize arrangements of lines

  • Concerning the maximum number of triangles in pairwise intersecting arrangements of pseudocircles, we show that p3 ≤

  • In this paper we report on some progress regarding conjectures involving numbers of triangles and digons in arrangements of pseudocircles

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Summary

Introduction

Arrangements of pseudocircles generalize arrangements of circles in the same vein as arrangements of pseudolines generalize arrangements of lines. Conjecture 3.7 from Grünbaum’s monograph [10] is: Every (not necessarily simple) digon-free arrangement of n pairwise intersecting pseudocircles has at least 2n − 4 triangles. Snoeyink and Hershberger [12] showed that the sweeping technique, which serves as an important tool for the study of arrangements of lines and pseudolines, can be adapted to work in the case of arrangements of pseudocircles They used sweeps to show that, in an intersecting arrangement of pseudocircles, every pseudocircle is incident to two cells which are digons or triangles on either side. Felsner and Kriegel [6] observed that the bound from [12] applies to non-simple intersecting digon-free arrangements and gave examples of arrangements showing that the bound is tight on this class for infinitely many values of n.

Intersecting Arrangements with Few Triangles
Intersecting Arrangements with Digons
Maximum Number of Triangles
Constructions Using Arrangements of Pseudolines
Visualization
Iterated Tutte Embeddings
Visualization Using Curves
A More General Representation

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