OFFSET
1,5
COMMENTS
A Venn diagram with n curves is a collection of n simple closed curves in the plane that intersect in only finitely many points and create exactly 2^n regions, one for every possible combination of being inside or outside of each of the n curves.
A Venn diagram is simple if at most two of the n curves intersect in any point.
For this counting sequence we consider two Venn diagrams equivalent if the differ only by mirroring and/or stereographic projection. These are also sometimes called Venn classes or spherical Venn diagrams in the literature.
The dual graphs are exactly all non-isomorphic planar spanning subgraphs of the n-dimensional hypercube in which every face has length 4, with the additional requirement that for every position i in {1,...,n} and every bit b in {0,1} the corresponding subgraph induced by all vertices x with x_i=b is connected.
REFERENCES
K. B. Chilakamarri, P. Hamburger, and R. E. Pippert, Analysis of Venn diagrams using cycles in graphs. Geom. Dedicata, 82(1-3):193-223, 2000.
P. Hamburger and R. E. Pippert, Simple, reducible Venn diagrams on five curves and Hamiltonian cycles. Geom. Dedicata, 68(3):245-262, 1997.
J. Venn. On the diagrammatic and mechanical representation of propositions and reasonings. Phil. Mag. S. 5., 9(59):1-18, 1880.
P. Winkler, Venn diagrams: some observations and an open problem, in Proceedings of the Fifteenth Southeastern Conference on Combinatorics, Graph Theory and Computing, pages 267-274, 1984.
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, thus, a(3)=1.
+-----------+
| |
| +---|-------+
| | | |
| +---|---|---+ |
| | | | | |
+---|---|---+ | |
| | | |
| +-------|---+
| |
+-----------+
The corresponding dual graph is the 3-cube.
CROSSREFS
KEYWORD
nonn,hard,more,bref
AUTHOR
Torsten Muetze, Oct 30 2025
STATUS
approved