the search for simple symmetric venn diagrams torsten mütze, eth zürich talk mainly based on...
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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.