Gábor Damásdi


Number of digons in arrangements of pairwise intersecting pseudo-circles. 

Given an arrangement of pairwise intersecting pseudo-circles, digons are the faces in this arrangement that have two edges.  A long-standing open conjecture of Branko Grünbaum from 1972 states that any simple arrangement of n pairwise intersecting pseudocircles in the plane can have at most 2n − 2 digons. Agarwal et al. proved this conjecture for arrangements of pairwise intersecting pseudocircles in which there is a common point surrounded by all pseudocircles. Recently, Felsner, Roch and Scheucher showed that Grünbaum’s conjecture is true for arrangements of pairwise intersecting pseudocircles in which there are three pseudocircles every pair of which create a digon. We show that the conjecture is true in general. Furthermore, we show that the digon-graph of the arrangement has a number of interesting properties. For example, even though it is not always planar, it is embeddable in the projective plane.  
Joint work with Eyal Ackerman, Eric Gottlieb, Balázs Keszegh, Rom Pinchasi
and Rebeka Raffay.