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

Variations and Extensions

Beyond what we've discussed in the preceding pages, there are many other aspects of Venn diagrams that have been considered in the literature. We mention some of these below.

A Venn diagram is *convex* if the interiors of all
its curves are convex.
Note that the regions of a Venn diagram can alway be made convex,
for example, by taking the Tutte embedding of the diagram.
Grünbaum defines a stronger notion of convexity by insisting
that not only are the curves convex, but also regions and the
complement of the external region; we call this condition
*strong convexity*.

Every monotone Venn diagram is isomorphic to a convex diagram via the following result of Bultena, Grünbaum, and Ruskey [BGR]. A FISC is a family of finitely intersecting simple closed curves in the plane, with the property that the intersection of the interiors of all the curves is not empty. Clearly, every Venn diagram is a FISC. A FISC is monotone if its directed dual graph has a unique source and a unique sink; this corresponds with our definition of a monotone Venn diagram.

**Theorem BGR**
A FISC has a convex drawing in the plane if and only if it is
monotone.
The survey article of Hamburger
[Ha98] contains a wealth
of material on geometric aspects of Venn diagrams, particularly
convexity issues.

Let *k*-gon designate any convex polygon with at most *k*
sides.
Define *N*(*k*) to be the maximum number
of curves in a Venn diagram whose curves are *k*-gons.
Grünbaum
[Gr75]
proves that *N*(*k*) is asympotically *lg*(*n*)
via a sqeezing argument, where *lg* denotes the base 2 logarithm.
The upper bound uses a construction very much like Edwards' general
construction, except that the circle is replaced by a square and the
*n*th curve is a 2^{n}-gon.
Also, Grünbaum
[Gr75, pg. 17]
states that "the construction we
used ... is a modification to convex polygons of the method
described in Moor [Mo]."

Define *K*(*n*) to be the smallest *k* for which there
exists a Venn diagram of *K*(*n*) curves, all of which
are *k*-gons.
Grünbaum
[Gr75]
contains a proof of the following theorem.

**Theorem G.** *K*(3) = *K*(4) = *K*(5) = 3, and
*K*(6) < 5.

The value of *K*(7) is unknown, but Grünbaum has constructed
and independent family of 7 hexagons.

Recently, Jeremy Carroll of HP labs (Bristol, England)
[Ca99] has solved
the problem of finding a collection of six triangles that
form a Venn diagram, thereby proving that *K*(6) = 3.
This was listed as an open problem in the first version of this
survey.

Six amazing triangles |

A Venn diagram is *exposed* if each of its curves touches the
outer face at some point of non-intersection.
Note, for example, that the 3-Venn diagram #3.2 is not exposed, even
though all curves have a point of intersection on the outer face.
Stated in terms of *n*-Venn graphs, being exposed is
the same as the condition
that the vertex corresponding to the outer face has degree *n*.
Every symmetric Venn diagram must be exposed (obvious) and
every convex Venn diagram must be exposed
([Gr92a]).
A Venn diagram has a *hidden curve* if it has a curve that
does not touch the outer region.
For example, in Venn's general construction, curves 4,5,...,*n*-1
are all hidden.
Chilakamarri, Hamburger, and Pippert
[CHP96b] gave an example, shown below,
of a simple 6-Venn diagram which is not in the same class as
any exposed diagram.
The hidden curve is the dotted green one.

All simple Venn diagrams with five or less curves are exposed, but with five curves there are 8 (5 reducible, 3 irreducible) that do not have a convex drawing in the plane [CHP97a].

Hamburger and Pippert [HP96a] proved that there are simple reducible Venn diagrams with 5 congruent ellipses, in spite of the fact that Venn had stated that there are no such diagrams. In fact, there are two of them, but they are in the same class.

Grünbaum [Gr75] mentions several
general geometric problems regarding *d*-dimensional diagrams where the
sets are required to be boundaries of *d*-polytopes or
convex regions.

If we relax the restriction that Venn diagrams meet in only finitely
many points, then we lose some of the nice graph theoretic
considerations (e.g., the dual graph is no longer a subgraph of the
hypercube), but do gain some flexibility.
Below we show a diagram made from three congruent curves, red, blue, and green.
Each curve is congruent and they are successively rotated by
**pi**/4 radians about the central point.

Such a construction is possible for any *n*, and is attributed to Peter
Shor in Knuth, Graham, and Patashnik [GKP].
The idea of the construction is to take a De Bruijn
cycle^{9}, *C*,
of length 2^{n}, together with a regular
2^{n}-gon inscribed in a circle.
Make a curve by successively choosing a circular arc for each 1
in *C* and a polygonal edge segment for each 0.
To obtain the other curves rotate this one about the central point by
2 *k* **pi** / 2^{n} radians for
*k* = 1,2,...,*n-*1.
The example shown above is created by starting with the De Bruijn cycle
*C* = 00011101.

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