1. The Search for Simple Symmetric Venn Diagrams Torsten Mütze, ETH Zürich Talk mainly based on [Griggs, Killian, Savage 2004] TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: A A A A AA A A

2. simplenon-simple Venn Diagrams A B C Def: n -Venn diagram - n Jordan curves in the plane - finitely many intersections - For each the region is nonempty and connected (=>2 n regions in total) n =3 Introduced by John Venn (1834–1923) for representing “propositions and reasonings”

3. Existence for all n ? Theorem (Venn 1880): There is a simple n -Venn diagram for every n. It won’t work with n circles: CnCn Proof: Induction over n with invariant: last curve added touches every region exactly once n =3 C n +1

4. Existence for all n n =4 n =5 n =6 What about diagrams that look “more nicely”?

5. Symmetric diagrams for all n ? Def: ( n -fold) symmetric Venn diagram Rotation of one curve around a fixed point yields all others Def: rank of a region = number of curves for which region is inside = number of 1 ’s in char. vect0r => regions of rank r n =3 n =5 n =7 Def: characteristic vector of a region region inside 001 100 010 101 011 110 111 000

6. Symmetric diagrams for all n ? Theorem (Henderson 1963): Necessary for the existence of a symmetric n -Venn diagram is that n is prime. n =4 regions of rank 2 6 is not divisible by 4 => no symmetric 4-Venn diagram Proof: is divisible by n for all iff n is prime (Leibniz)

7. Symmetric diagrams for all prime n non-simple Theorem (Griggs, Killian, Savage 2004): If n is prime, then there is a symmetric n -Venn diagram. n =5 n =11

8. 100 010 110 000 G‘G‘ 001 101 111 011 Basic observations any n prime n Forget about symmetry for the moment (following holds for any n) G n =4 n =3 G View Venn diagram as (multi)graph G Observation: Geometric dual G ‘ is a subgraph of Q n 001 100 010 101 011 110 111 000 G ‘= Q 3 G ‘= Q 4 minus 4 edges Idea: Reverse the construction Want: Subgraph G ‘ of Q n that is planar spanning dual edges of the i -edges in G ‘ form a cycle in G i -edges form bond ( G ‘ minus i -edges has exactly two components) 3-edges

9. Basic observations any n Want: Subgraph of Q n that is planar, spanning, i -edges form bond => dual is a Venn diagram Want: Subgraph of Q n that is planar, spanning, monotone Lemma: monotone => i -edges form bond Proof: View Q n as boolean lattice 1110 1101 1011 0111 1111 1000 0100 0010 0001 0000 1001 0101 0011 1100 1010 0110 Q4Q4 1110 0111 1111 1000 0100 0001 0000 0011 1010 0110 1110 1101 1011 0111 1111 1000 0100 0010 0001 0000 1001 0101 0011 1100 1010 0110 3-edges 1100110 1101110 0100110 Def: monotone subgraph of Q n every vertex has a neighbor with 0 1, and one with 1 0 (except 0 n and 1 n )

10. any n Def: symmetric chain in Q n QnQn 0n0n 1n1n chain symmetric chain Theorem (Greene, Kleitman 1976): Q n has a decomposition into symmetric chains. Q4Q4 Symmetric chain decomposition Greene-Kleitman decomposition + extra edges => dual is a Venn diagram

11. How to achieve symmetry Idea: Work within “1/ n -th” of Q n to obtain “1/ n -th” of Venn diagram, then rotate Now suppose n is prime Prime n => natural partition of Q n into n symmetric classes Def: necklace = set of all n -bit strings that differ by rotation { 11000, 10001, 00011, 00110, 01100 } n =5: { 11010, 10101, 01011, 10110, 01101 } 2 necklaces Observation: Prime n => each necklace has exactly n elements (except { 0 n } and { 1 n }) Want: Suitable set R n of necklace representatives + a planar, spanning, monotone subgraph of Q n [ R n ] (via SCD) => symmetric Venn diagram prime n n =5

12. Necklaces in action Want: Suitable set R n of necklace representatives + a planar, spanning, monotone subgraph of Q n [ R n ] (via SCD) prime n n =5: n elements per necklace { 0 n }, { 1 n } number of necklaces of size n i -edge becomes ( i -1)-edge in the next slice Q5[R5]Q5[R5] { 11010, 10101, 01011, 10110, 01101 } 11111 00000 10000 10110 11110 11000 10100 11100 SCD + extra edges 11111 00000 10000 10100 11100 10110 11110 11000

13. Symmetric chain decomposition of Q n any n Theorem (Greene, Kleitman 1976): Q n has a decomposition into symmetric chains. Proof: Parentheses matching: 0 = ( 1 = ) match parentheses in the natural way from left to right Observations: unmatched ‘s are left to unmatched ‘s 1 0 flipping rightmost or leftmost does not change matched pairs 1 0 Chains uniquely identified by matched pairs Repeat this flipping operation => symmetric chain decomposition 10 0 0 1 1 0010 0 00 0 0 1 1 0010 0 11 0 0 1 1 1011 1 11 0 0 1 1 0010 0 11 0 0 1 1 1011 0 11 0 0 1 1 1010 0 Q 11 0 0 1 1 01 1110 0

14. Join each chain to its parent chain Adding the extra edges any n 10 0 0 1 1 0010 0 00 0 0 1 1 0010 0 11 0 0 1 1 1011 1 11 0 0 1 1 0010 0 11 0 0 1 1 1011 0 Q 11 Def: parent chain of a chain = flip the in the rightmost matched pair 1 1110 0 0 0 1 1 01 00 0 0 1 1 0000 0 chain 11 0 0 1 1 0000 0 10 0 0 1 1 0000 0 11 0 0 1 1 1111 0 11 0 0 1 1 1000 0 11 0 0 1 1 1110 0 1110 0 0 0 1 1 10 11 0 0 1 1 1111 1 parent chain

15. Adding the extra edges Q4Q4 Embed parent chain, then left children before right children any n 1 2 3 4 5 6 1 2 3 4 5 6 parent chain => planar, spanning, monotone subgraph of Q n Join each chain to its parent chain

16. Symmetric chain decomposition of Q n [ R n ] Main contribution of [Griggs, Killian, Savage 2004] 11001100100 (4,4,3) 10011001001 00110010011 01100100110 11001001100 10010011001 00100110011 01001100110 10011001100 00110011001 01100110010 (3,4,4) (4,3,4) (∞)(∞) (∞)(∞) (∞)(∞) (∞)(∞) (∞)(∞) (∞)(∞) (∞)(∞) (∞)(∞) block code necklace all finite block codes differ by rotation 11001100100 Def: block code of a 0 - 1 - string (4, 4, 3) n =11: 0xxxxxxxxxx xxxxxxxxxx1 (∞)(∞) (∞)(∞) n prime: no two elements with same finite block code From each necklace select element with lexicographically smallest block code as representative => R n Observations: In each necklace (except { 0 n } and { 1 n }) at least one finite block code prime n

17. => symmetric chain decomposition of Q n [ R n ] Symmetric chain decomposition of Q n [ R n ] prime n Observation: Block codes within Greene-Kleitman chain do not change (except ( ∞ ) at both ends) => chain with one element from R n contains only elements from R n Add extra edges between chains to obtain planar, spanning, monotone subgraph of Q n [ R n ] 00 0 1 0 0 0110 0 11 0 1 1 0 0111 1 11 0 1 1 0 0111 0 1100 0 0 1 1 0 11 10 0 1 0 0 0110 011 0 1 0 0 0110 0 (3,4,4) Q 11 [ R 11 ] block code (3,4,4) (∞)(∞) (∞)(∞)

18. Making the diagram simpler prime n # vertices in the resulting Venn diagram = # faces of the subgraph of Q n = # chains in the SCD = # vertices in a simple Venn diagram = 2 n -2 => increase the number of vertices to at least (2 n -2)/2 Observation: Faces between neighboring chains can be quadrangulated Q7[R7]Q7[R7] Question: Is there a simple symmetric n-Venn diagram for prime n ?

19. Thank you! Questions?

20. References Jerrold Griggs, Charles E. Killian, and Carla D. Savage. Venn diagrams and symmetric chain decompositions in the Boolean lattice. Electron. J. Combin., 11:Research Paper 2, 30 pp. (electronic), 2004. [Griggs, Killian, Savage 2004] Frank Ruskey. A survey of Venn diagrams. Electron. J. Combin., 4(1):Dynamic Survey 5 (electronic), 2001. Charles E. Killian, Frank Ruskey, Carla D. Savage, and Mark Weston. Half- simple symmetric Venn diagrams. Electron. J. Combin., 11:Research Paper 86, 22 pp. (electronic), 2004.