Venn diagrams with few intersections
Date
1998
Authors
Bultena, Albertha
Journal Title
Journal ISSN
Volume Title
Publisher
Abstract
The n-Venn diagram is a collection of simple closed curves in the plane, intersecting only at point. The curves divide the plane into 2n open connected regions. Further, each region must contain a unique set of interiors of the curve. A region's weight is the number of curves that contain it. A monotone Venn diagram with n curves has the property that every region with weight k, where 1 <k <n, is adjacent to at least one region with weight k - 1 and at least one region with weight k + 1. An n-Venn diagram can be interpreted as a planar graph in which the intersection points of the curves are the vertices. We show that each monotone Venn diagram has at least ____ vertices and that this bound can be attained for all n > 1. For general Venn diagrams, the number of vertices is at least ______. Examples are given that demonstrate that this bound can be attained for 1 < n < 7.