THE ELECTRONIC JOURNAL OF COMBINATORICS (ed. March 2001), DS #5.

# Venn Diagram Survey Open Problems

## Open problems related to Venn diagrams.

• Find a symmetric Venn diagram for n = 13, or prove that no such diagram exists.

• How many symmetric Venn diagrams are there for n = 5? There is only one symmetric simple 5-Venn diagram.

• How many symmetric Venn diagrams are there for n = 7? How many of these are simple?

• Is it true that every simple Venn diagram of n curves can be extended to a simple Venn diagram of n+1 curves by the addition of a suitable curve? [That this is true is a conjecture of Winkler [Wi]. This was proven to be true for not necessarily simple Venn diagrams by Chilakamarri, Hamburger, and Pippert [CHP96].] Equivalently: Is every planar dual graph of a simple Venn diagram Hamiltonian?

• Find a 6-Venn diagram in which each curve is a rectangle. This problem is from Grünbaum [Gr84b]. His other problem of finding a 6-Venn diagram in which each curve is a triangle has been answered affirmatively. See the six amazing triangles.

• Find a 6-Venn diagram made from equilateral triangles. Even the problem of finding a 6-Independent family made from equilateral triangles is open. This problem from Grünbaum [Gr75].

• There is no Venn diagram of 8 convex quadrangles. Does there exist a Venn diagram of 8 quadrangles not required to be convex, or of 8 pentagons? These problems are due to Grünbaum.

• Find a Brunnian link whose minimal projection is a symmetric Venn diagram of order 7 or prove that no such link exists.

• Find a symmetric k-fold (k > 2) Venn diagram on the sphere or prove that no such diagram exists. For n = 2 there are diagrams that, in the plane, have rotational symmetry, and and examples that have mirror-image symmetry (these examples due to Branko Grünbaum).

• For n > 2, find a "symmetric" diagram on the sphere where the group of symmetries is different than those used in the section on symmetric Venn diagrams. In that section the group of symmetries consisted of rotations about a single axis, or rotations and a flip about that axis. In other words, the problem is: Are there other groups of isometries that act transitively on the curves of some Venn diagram on the sphere?

• The least number of vertices in a monotone Venn diagram is known to be C(n,n/2); what is the least number of vertices in a general Venn diagram? The lower bound of ceiling((2n-2)/(n-1)) is known to be achieveable for n < 8, so the first open case is n = 8. It is not known whether there is a rigid minimum vertex Venn diagram for n = 7.

• Are there rigid* monotone n-Venn diagrams with C(n,n/2) vertices? The constructions of [BGR] have C(n,n/2) vertices but have many separable vertices.

• The dual of a simple Venn diagram is a maximal planar spanning subgraph of the hypercube. Does every maximal spanning subgraph of the hypercube occur as the Venn graph of some Venn diagram?

• For every n, is there a convex Venn diagram made from congruent curves?

• Over all n-Venn diagrams, which requires the greatest number of straight line segments to draw it?

 THE ELECTRONIC JOURNAL OF COMBINATORICS (ed. March 2001), DS #5.