Which n-Venn diagrams can be drawn with convex k-gons?
Abstract
We establish a new lower bound for the number of sides required for the component curves of simple Venn diagrams made from polygons. Specifically, for any n-Venn diagram of convex k-gons, we prove that k >= (2^n - 2 - n) / (n (n-2)). In the process we prove that Venn diagrams of seven curves, simple or not, cannot be formed from triangles. We then give an example achieving the new lower bound of a (simple, symmetric) Venn diagram of seven quadrilaterals. Previously Grunbaum had constructed a 7-Venn diagram of non-convex 5-gons [``Venn Diagrams II'', Geombinatorics 2:25-31, 1992].
Cite
@article{arxiv.cs/0512001,
title = {Which n-Venn diagrams can be drawn with convex k-gons?},
author = {Jeremy Carroll and Frank Ruskey and Mark Weston},
journal= {arXiv preprint arXiv:cs/0512001},
year = {2007}
}
Comments
10 pages, 3 figures. To be published in Proceedings of the Second International Workshop on Euler Diagrams (Euler 2005), Electronic Notes in Theoretical Computer Science