OFFSET
1,5
COMMENTS
See A386795 for the definition of simple Venn diagrams.
A k-region in a Venn diagram with n curves is a region that is inside of exactly k of the curves and outside of the remaining n-k curves. A monotone Venn diagram is one in which for every 0<k<n, every k-region is adjacent to both a (k-1)-region and a (k+1)-region. Monotone Venn diagrams are precisely those can be drawn by curves that have a convex shape.
The dual graphs are exactly all non-isomorphic planar spanning subgraphs of the n-dimensional hypercube in which all vertices of weight k are embedded at vertical height k, for k=0,...,n, and every face has length 4 and spans three consecutive levels k-1,k,k+1. The weight of a vertex is the number of 1s in it.
REFERENCES
B. Bultena, B. Grünbaum, and F. Ruskey, Convex drawings of intersecting families of simple closed curves, in Proceedings of the 11th Canadian Conference on Computational Geometry, 1999.
LINKS
Sofia Brenner, Petr Gregor, Torsten Mütze, and Francesco Verciani, On minimum Venn diagrams, arXiv:2511.09230 [math.CO], 2025.
Sofia Brenner, Linda Kleist, Torsten Mütze, Christian Rieck, and Francesco Verciani, Counterexamples to two conjectures on Venn diagrams, arXiv:2503.18554 [math.CO], 2025.
Frank Ruskey and Mark Weston, A survey of Venn diagrams, Electron. J. Combin., Dynamic Survey 5, 1997.
EXAMPLE
For n=3 curves, there is only the simple Venn diagram shown in the following figure, and it is monotone, thus, a(3)=1.
+-----------+
| |
| +---|-------+
| | | |
| +---|---|---+ |
| | | | | |
+---|---|---+ | |
| | | |
| +-------|---+
| |
+-----------+
The corresponding dual graph is the 3-cube.
CROSSREFS
KEYWORD
nonn,hard,bref
AUTHOR
Torsten Muetze, Oct 30 2025
STATUS
approved