Venn Diagrams with Few Vertices

An n-Venn diagram is a collection of n finitely-intersecting simple closed curves in the plane, such that each of the 2n sets X1 ∩X2 ∩· · ·∩Xn, where each Xi is the open interior or exterior of the i-th curve, is a non-empty connected region. The weight of a region is the number of curves that contain it. A region of weight k is a k-region. A monotone Venn diagram with n curves has the property that every k-region, where 0 < k < n, is adjacent to at least one (k −1)-region and at least one (k +1)-region. Monotone diagrams are precisely those that can be drawn with all curves convex. An n-Venn diagram can be interpreted as a planar graph in which the intersection points of the curves are the vertices. For general Venn diagrams, the number of vertices is at least 2n−2 n−1 . Examples are given that demonstrate that this bound can be attained for 1 < n ≤ 7. We show that each monotone Venn diagram has at least n n/2 vertices, and that this lower bound can be

• Type: Journal
• Year: 1998
• Where: COMBINATORICS
• Authors: Bette Bultena, Frank Ruskey
• Keywords: COMBINATORICS 1998, Monotone Venn Diagram, N-Venn Diagram, Non-empty Connected Region