The Search for Simple Symmetric Venn Diagrams

Torsten Mütze, ETH ZürichTalk mainly based on [Griggs, Killian, Savage 2004]

simple non-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 (=>2n 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:

Cn

• Proof: Induction over n with invariant:last curve added touches every region exactly once

n=3 Cn+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 regionregion inside

001

100

010

101

011

110

111

000

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

100

010

110

000

G‘001

101

111

011

Basic observations

any n

prime n

• Forget about symmetry for the moment (following holds for any n)

G

n=4n=3

G

• View Venn diagram as (multi)graph G

• Observation: Geometric dual G‘ is a subgraph of Qn

001

100

010

101

011

110

111

000G‘=Q3

G‘=Q4minus 4 edges• Idea: Reverse the construction• Want: Subgraph G‘ of Qn 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 Qn that is planar, spanning, i-edges form bond

=> dual is a Venn diagram

• Want: Subgraph of Qn that isplanar, spanning, monotone

• Lemma: monotone => i-edges form bond

• Proof:

• View Qn as boolean lattice

1110110110110111

1111

1000010000100001

0000

100101010011 110010100110

Q411100111

1111

100001000001

0000

0011 10100110

1110110110110111

1111

1000010000100001

0000

100101010011 110010100110

3-edges

1100110

1101110

0100110

• Def: monotone subgraph of Qn

every vertex has a neighbor with 0 1,and one with 1 0 (except 0n and 1n)

any n

• Def: symmetric chain in Qn

Qn

0n

1n

chainsymmetricchain

• Theorem (Greene, Kleitman 1976): Qn has a decomposition into symmetric chains.

Q4

Symmetric chaindecomposition

• Greene-Kleitman decomposition + extra edges => dual is a Venn diagram

How to achieve symmetry

• Idea: Work within “1/n-th” of Qn to obtain “1/n-th” of Venn diagram, then rotate

• Now suppose n is prime

• Prime n => natural partition of Qn 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 {0n} and {1n})

• Want: Suitable set Rn of necklace representatives + a planar, spanning, monotone subgraph of Qn[Rn] (via SCD)

=> symmetric Venn diagram

prime n

n=5

Necklaces in action• Want: Suitable set Rn of necklace representatives

+ a planar, spanning, monotone subgraph of Qn[Rn] (via SCD)

prime n

n=5: n elements per necklace

{0n}, {1n}

number of necklaces of size n

i-edge becomes(i-1)-edge in the

next sliceQ5[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

Symmetric chain decomposition of Qn

any n

• Theorem (Greene, Kleitman 1976): Qn has a decomposition into symmetric chains.

• Proof: Parentheses matching: 0 = ( 1 = )match parentheses in the natural wayfrom left to right

• Observations:• unmatched ‘s are left

to unmatched ‘s10

• flipping rightmost or leftmost does not change matched pairs

1 0

• Chains uniquely identified by matched pairs

• Repeat this flipping operation => symmetric chain decomposition

1000110010 0

00001100100

11001110111

1100110010 0

1100111011 0

11001110100Q11 0011 0111 1 00

• Join each chain to its parent chain

Adding the extra edges

any n

10001100100

00001100100

11001110111

1100110010 0

1100111011 0

Q11

• Def: parent chain of a chain = flip the in the rightmost matched pair1

11 1 000011 01

00001100000

chain

11001100000

10001100000

11001111110

11001110000

1100111110 0

11 1 000011 10

11001111111

pare

nt

chain

Adding the extra edges

Q4

• Embed parent chain, then left children before right children

any n

1

2

3

4

5

6

1

2

3

4

5

6

parentchain

=> planar, spanning, monotone subgraph of Qn

• Join each chain to its parent chain

Symmetric chain decomposition of Qn[Rn]

• Main contribution of [Griggs, Killian, Savage 2004]

11001100100 (4,4,3)10011001001001100100110110010011011001001100100100110010010011001101001100110100110011000011001100101100110010

(3,4,4)

(4,3,4)

(∞) (∞) (∞)

(∞) (∞) (∞)

(∞) (∞)

block code

neck

lace

• all finite block codes differ by rotation

11001100100

• Def: block code of a 0-1-string

(4 , 4 , 3)

n=11: 0xxxxxxxxxxxxxxxxxxxx1

(∞) (∞)

• n prime: no two elements with same finite block code

• From each necklace select element with lexicographically smallest block code as representative => Rn

• Observations: In each necklace (except {0n} and {1n})• at least one finite block code

prime n

=> symmetric chain decomposition of Qn[Rn]

Symmetric chain decomposition of Qn[Rn]

prime n

• Observation: Block codes within Greene-Kleitman chain do not change (except (∞) at both ends)

=> chain with one element from Rn contains only elements from Rn

• Add extra edges between chains to obtainplanar, spanning, monotonesubgraph of Qn[Rn]

00010001100

11011001111

1101100111 0

11 0 000110 11

10010001100

11010001100 (3,4,4)Q11[R11]

block code

(3,4,4)

(3,4,4)

(∞)

(∞)

(3,4,4)

Making the diagram simpler

prime n

• # vertices in the resulting Venn diagram = # faces of the subgraph of Qn

= # chains in the SCD =

• # vertices in a simple Venn diagram = 2n-2

=> increase the number of vertices to at least (2n-2)/2

• Observation: Faces between neighboring chains can be quadrangulated

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), 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.