2024/03/08 by Pinchasi, Rom
#Combinatorics (math.CO) #FOS: Mathematics
paper · doi:10.48550/arxiv.2403.05270
Let \F be a family of n pairwise intersecting circles in the plane. We show that the number of lenses, that is convex digons, in the arrangement induced by \F is at most 2n-2. This bound is tight. Furthermore, if no two circles in \F touch, then the geometric graph G on the set of centers of the circles in \F whose edges correspond to the lenses generated by \F does not contain pairs of avoiding edges. That is, G does not contain pairs of edges that are opposite edges in a convex quadrilateral. Such graphs are known to have at most 2n-2 edges.