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

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”

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

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

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

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)

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

G‘G‘ 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 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

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 Q4Q edges 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 )

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

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, } n =5: { 11010, 10101, 01011, 10110, } 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

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, } SCD + extra edges

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 Q

Join each chain to its parent chain Adding the extra edges any n Q 11 Def: parent chain of a chain = flip the in the rightmost matched pair chain parent chain

Adding the extra edges Q4Q4 Embed parent chain, then left children before right children any n parent chain => planar, spanning, monotone subgraph of Q n Join each chain to its parent chain

Symmetric chain decomposition of Q n [ R n ] Main contribution of [Griggs, Killian, Savage 2004] (4,4,3) (3,4,4) (4,3,4) (∞)(∞) (∞)(∞) (∞)(∞) (∞)(∞) (∞)(∞) (∞)(∞) (∞)(∞) (∞)(∞) block code necklace all finite block codes differ by rotation Def: block code of a 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

=> 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 ] (3,4,4) Q 11 [ R 11 ] block code (3,4,4) (∞)(∞) (∞)(∞)

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 ?

Thank you! Questions?

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), [Griggs, Killian, Savage 2004] Frank Ruskey. A survey of Venn diagrams. Electron. J. Combin., 4(1):Dynamic Survey 5 (electronic), 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.