 
Summary: On The Complexity of Arrangements of
Circles in the Plane
Noga Alon
Hagit Last
Rom Pinchasi§
Micha Sharir¶
February 22, 2002
Abstract
Continuing and extending the analysis in a previous paper [9], we establish
several combinatorial results on the complexity of arrangements of circles in the
plane. The main results are a collection of partial solutions to the conjecture
that (a) any arrangement of unit circles with at least one intersecting pair has
a vertex incident to at most 3 circles, and (b) any arrangement of circles of
arbitrary radii with at least one intersecting pair has a vertex incident to at
most 3 circles, provided the number of intersecting pairs is significantly larger
than the number of circles times the maximum cardinality of a subset of circles
incident to a fixed pair of points.
1 Introduction
In this paper we study the combinatorial complexity of arrangements of circles in the
plane. The main motivation for our study is the following conjecture, some of whose
