Definitions
Disjoint edges in complete topological graphs
Andrew Suk
MIT
August 19, 2012
Andrew SukMIT
Disjoint edges in complete topological graphs
Definitions
Main problem
Problem: Given a collection of objects C in the plane, what is the
size of the largest subcollection of pairwise disjoint objects?
Andrew SukMIT
Disjoint edges in complete topological graphs
Definitions
Main problem
Problem: Given a collection of objects C in the plane, what is the
size of the largest subcollection of pairwise disjoint objects?
Andrew SukMIT
Disjoint edges in complete topological graphs
Definitions
Motivation
Map labeling example. Computer contains the names of every
street, river, park, shop, etc.
Map near NYU
Starbucks
La Guardia
PLace
Schwartz Plaza
Greene Street
Bobst Library
Washington Square Park
Courant
Institute
Think
Coffee
Campus
DogEatery
Park Student
SternServices
3rd StreetNYU
Andrew SukMIT
Disjoint edges in complete topological graphs
Definitions
Motivation
Map labeling example. Computer contains the names of every
street, river, park, shop, etc.
Map near NYU
Starbucks
Greene Street
Bobst Library
Courant Institute
Campus Eatery
3rd Street
Andrew SukMIT
Disjoint edges in complete topological graphs
Definitions
We will consider the case when C is the collection of edges in a
simple topological graph.
Andrew SukMIT
Disjoint edges in complete topological graphs
Definitions
Definition
A topological graph is a graph drawn in the plane with vertices
represented by points and edges represented by curves connecting
the corresponding points. A topological graph is simple if every
pair of its edges intersect at most once.
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Andrew SukMIT
Disjoint edges in complete topological graphs
Definitions
Definition
A topological graph is a graph drawn in the plane with vertices
represented by points and edges represented by curves connecting
the corresponding points. A topological graph is simple if every
pair of its edges intersect at most once.
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Andrew SukMIT
Disjoint edges in complete topological graphs
Definitions
We will only consider simple topological graphs.
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Andrew SukMIT
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Disjoint edges in complete topological graphs
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Definitions
Three problems in topological graph theory.
Problem 1: Thrackle conjecture.
Problem (Conway)
Does every n-vertex simple topological graph with |E (G )| > n have
two disjoint edges?
Fulek and Pach 2010: |E (G )| ≥ 1.43n.
Best known 1.43n by Fulek and Pach, 2010.
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Andrew SukMIT
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Disjoint edges in complete topological graphs
Definitions
Three problems in topological graph theory.
Problem 1: Thrackle conjecture.
Problem (Conway)
Does every n-vertex simple topological graph with |E (G )| > n have
two disjoint edges?
Fulek and Pach 2010: |E (G )| ≥ 1.43n.
Best known 1.43n by Fulek and Pach, 2010.
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Andrew SukMIT
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Disjoint edges in complete topological graphs
Definitions
Generalization.
Theorem (Pach and Tóth, 2005)
Every n-vertex simple topological graph with no k pairwise disjoint
edges, has at most Ck n log5k−10 n edges.
Conjecture to be at most O(n) (for fixed k). By solving for k in
Ck n log5k−10 n = n2 .
Corollary (Pach and Tóth, 2005)
Every complete n-vertex simple topological graph has at least
Ω(log n/ log log n) pairwise disjoint edges.
Andrew SukMIT
Disjoint edges in complete topological graphs
Definitions
Conjecture (Pach and Tóth)
There exists a constant δ, such that every complete n-vertex
simple topological graph has at least Ω(nδ ) pairwise disjoint edges.
Andrew SukMIT
Disjoint edges in complete topological graphs
Definitions
History
Pairwise disjoint edges in complete n-vertex simple topological
graphs:
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Ω(log1/6 n), Pach, Solymosi, Tóth, 2001.
2
Ω(log n/ log log n), Pach and Tóth, 2005.
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Ω(log1+ǫ n), Fox and Sudakov, 2008.
Note ǫ ≈ 1/50. All results are slightly stronger statements.
Andrew SukMIT
Disjoint edges in complete topological graphs
Definitions
Main result
Theorem (Suk, 2011)
Every complete n-vertex simple topological graph has at least
Ω(n1/3 ) pairwise disjoint edges.
Andrew SukMIT
Disjoint edges in complete topological graphs
Definitions
Clearly the simple condition is required.
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Disjoint edges in complete topological graphs
Definitions
Clearly the simple condition is required for this problem.
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Disjoint edges in complete topological graphs
Definitions
Clearly the simple condition is required for this problem.
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Disjoint edges in complete topological graphs
Definitions
Main result
Theorem (Suk, 2011)
Every complete n-vertex simple topological graph has at least
Ω(n1/3 ) pairwise disjoint edges.
Andrew SukMIT
Disjoint edges in complete topological graphs
Definitions
combinatorial tools
Let F be a set system with ground set X .
Definition (Dual shatter function)
∗ (m), is defined to be the maximum
The dual shatter function πF
number of equivalence classes on X , defined by an m-element
subfamily of F.
For m sets S1 , S2 , ..., Sm , x ∼ y if BOTH x, y are in exactly the
same sets among S1 , ..., Sm (i.e. no set Si contains x and not y or
vise versa).
∗ (m) is the number of nonempty cells in the Venn diagram
I.e. πF
of m sets of F.
Andrew SukMIT
Disjoint edges in complete topological graphs
Definitions
Example: X is a set of n points in the plane, F is the set of all
halfplanes.
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∗ (m) = O(m2 ).
πF
Andrew SukMIT
Disjoint edges in complete topological graphs
Definitions
Example: X is a set of n points in the plane, F is the set of all
halfplanes.
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∗ (m) = O(m2 ).
πF
Andrew SukMIT
Disjoint edges in complete topological graphs
Definitions
Example: X is a set of n points in the plane, F is the set of all
halfplanes.
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∗ (m) = O(m2 ).
πF
Andrew SukMIT
Disjoint edges in complete topological graphs
Definitions
Example: X is a set of n points in the plane, F is the set of all
halfplanes.
111111111
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∗ (m) = O(m2 ).
πF
Andrew SukMIT
Disjoint edges in complete topological graphs
Definitions
definition
A set S ∈ F stabs the pair (of vertices) x, y if |S ∩ {x, y }| = 1.
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Andrew SukMIT
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Disjoint edges in complete topological graphs
Definitions
definition
A set S ∈ F stabs the pair (of vertices) x, y if |S ∩ {x, y }| = 1.
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Andrew SukMIT
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Disjoint edges in complete topological graphs
Definitions
definition
A set S ∈ F stabs the pair (of vertices) x, y if |S ∩ {x, y }| = 1.
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Andrew SukMIT
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Disjoint edges in complete topological graphs
Definitions
definition
A set S ∈ F stabs the pair (of vertices) x, y if |S ∩ {x, y }| = 1.
S
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Andrew SukMIT
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Disjoint edges in complete topological graphs
Definitions
Main tool
Theorem (Matching theorem, Chazelle and Welzl, 1989)
Let F be a set system on an n element point set X (n is even),
∗ (m) ≤ O(md ). Then there exists a perfect matching
such that πF
M on X such that each set in F stabs at most O(n1−1/d )
members in M.
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Andrew SukMIT
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Disjoint edges in complete topological graphs
Definitions
Main tool
Theorem (Chazelle and Welzl, 1989)
Let F be a set system on an n-point set X (n is even), such that
∗ (m) ≤ O(md ). Then there exists a perfect matching M on X
πF
such that each set in F stabs at most O(n1−1/d ) members in M.
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M = {(x1 , y1 ), (x2 , y2 ), ...., (xn/2 , yn/2 )}.
Andrew SukMIT
Disjoint edges in complete topological graphs
Definitions
Main tool
Theorem (Chazelle and Welzl, 1989)
Let F be a set system on an n-point set X (n is even), such that
∗ (m) ≤ O(md ). Then there exists a perfect matching M on X
πF
such that each set in F stabs at most O(n1−1/d ) members in M.
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M = {(x1 , y1 ), (x2 , y2 ), ...., (xn/2 , yn/2 )}.
Andrew SukMIT
Disjoint edges in complete topological graphs
Definitions
Main tool
Theorem (Matching Lemma, Chazelle and Welzl 1989)
Let F be a set system on an n-point set X (n is even), such that
∗ (m) ≤ O(md ). Then there exists a perfect matching M on X
πF
such that each set in F stabs at most O(n1−1/d ) members in M.
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M = {(x1 , y1 ), (x2 , y2 ), ...., (xn/2 , yn/2 )}.
Andrew SukMIT
Disjoint edges in complete topological graphs
Definitions
Sketch of proof
Theorem (Suk, 2011)
Every complete n-vertex simple topological graph has at least
Ω(n1/3 ) pairwise disjoint edges.
Kn+1
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Andrew SukMIT
Disjoint edges in complete topological graphs
Definitions
Sketch of proof
Theorem (Suk, 2011)
Every complete n-vertex simple topological graph has at least
Ω(n1/3 ) pairwise disjoint edges.
Kn+1
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Disjoint edges in complete topological graphs
Definitions
Define F1 =
S
Si ,j , where Si ,j is the set of vertices inside
1≤i <j≤n
triangle v0 , vi , vj .
v0
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S3,9 = {v1 , v5 , v7 }
Andrew SukMIT
Disjoint edges in complete topological graphs
Definitions
Define F1 =
S
Si ,j , where Si ,j is the set of vertices inside
1≤i <j≤n
triangle v0 , vi , vj .
v0
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S3,9 = {v1 , v5 , v7 }, S2,11 = {v1 , v5 , v9 }
Andrew SukMIT
Disjoint edges in complete topological graphs
Definitions
Define F1 =
S
Si ,j , where Si ,j is the set of vertices inside
1≤i <j≤n
triangle v0 , vi , vj .
v0
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S3,9 = {v1 , v5 , v7 }, S2,11 = {v1 , v5 , v9 }, S5,11 = {v9 }.
Andrew SukMIT
Disjoint edges in complete topological graphs
Definitions
F1 is not ”complicated”.
Lemma
∗ (m) ≤ O(m2 ).
πF
1
Proof: Basically m ”triangles” divides the plane into at most
O(m2 ) regions. Proof is by induction on m.
Andrew SukMIT
Disjoint edges in complete topological graphs
Definitions
S
Define set system F2 =
1≤i <j≤n
Si′,j , where vk ∈ Si′,j if topological
edges v0 vk and vi vj cross.
v0
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Andrew SukMIT
Disjoint edges in complete topological graphs
Definitions
Define set system F2 =
S
1≤i <j≤n
Si′,j , where vk ∈ Si′,j if topological
edges v0 vk and vi vj cross.
v0
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′ = {v , v , v }.
S3,7
2 6 5
Andrew SukMIT
Disjoint edges in complete topological graphs
Definitions
S
Define set system F2 =
1≤i <j≤n
Si′,j , where vk ∈ Si′,j if topological
edges v0 vk and vi vj cross.
v0
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′ = {v , v , v }, S ′ =?.
S3,7
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2,9
Andrew SukMIT
Disjoint edges in complete topological graphs
Definitions
Define set system F2 =
S
1≤i <j≤n
Si′,j , where vk ∈ Si′,j if topological
edges v0 vk and vi vj cross.
v0
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v1211
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′ = {v , v , v }, S ′ = {v , v , v , v }.
S3,7
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1 3 6 12
2,9
Andrew SukMIT
Disjoint edges in complete topological graphs
Definitions
Again, F2 is not ”complicated”. Set F = F1 ∪ F2 . One can show
Lemma
∗ (m) = O(m3 ).
πF
Andrew SukMIT
Disjoint edges in complete topological graphs
Definitions
∗ (m) = O(m3 ).
πF
By the matching lemma (Chazelle and Welzl), there is a perfect
matching M such that each set in F = F1 ∪ F2 stabs at most
O(n2/3 ) members in M. Recall |M| = n/2.
v011
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Andrew SukMIT
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Disjoint edges in complete topological graphs
Definitions
∗ (m) = O(m3 ).
πF
By the matching lemma (Chazelle and Welzl), there is a perfect
matching M such that each set in F = F1 ∪ F2 stabs at most
O(n2/3 ) members in M. Recall |M| = n/2.
v011
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Andrew SukMIT
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Disjoint edges in complete topological graphs
Definitions
∗ (m) = O(m3 ).
πF
By the matching lemma (Chazelle and Welzl), there is a perfect
matching M such that each set in F = F1 ∪ F2 stabs at most
O(n2/3 ) members in M. Recall |M| = n/2.
v011
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Andrew SukMIT
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Disjoint edges in complete topological graphs
Definitions
Auxiliary graph G , where V (G ) = M and vi vj → vk vl if Si ,j or Si′,j
stabs vk vl .
v011
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Andrew SukMIT
Disjoint edges in complete topological graphs
Definitions
Auxiliary graph G , where V (G ) = M and vi vj → vk vl if Si ,j or Si′,j
stabs vk vl .
v011
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Andrew SukMIT
Disjoint edges in complete topological graphs
Definitions
Auxiliary graph G , where V (G ) = M and vi vj → vk vl if Si ,j or Si′,j
stabs vk vl .
v011
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vk
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Andrew SukMIT
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vj 11
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Disjoint edges in complete topological graphs
Definitions
Auxiliary graph G , where V (G ) = M and vi vj → vk vl if Si ,j or Si′,j
stabs vk vl .
v011
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Si ,j and Si′,j stabs (in total) at most O(n2/3 ) members in
M = V (G ). |E (G )| ≤ O(n5/3 ).
Andrew SukMIT
Disjoint edges in complete topological graphs
Definitions
|E (G )| ≤ O(n5/3 ), by Turán, G contains an independent set of
size Ω(n1/3 ).
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Andrew SukMIT
Disjoint edges in complete topological graphs
Definitions
|E (G )| ≤ O(n5/3 ), by Turán, G contains an independent set of
size Ω(n1/3 ).
v011
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Claim!
Andrew SukMIT
Disjoint edges in complete topological graphs
Definitions
|E (G )| ≤ O(n5/3 ), by Turán, G contains an independent set of
size Ω(n1/3 ).
v011
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vk
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Claim!
Andrew SukMIT
Disjoint edges in complete topological graphs
Definitions
Since Si ,j does NOT stab vk vl
v0
v0
1
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0
1
1
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0
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vl
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vi
1
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vj
vk
vk
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v1
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vi
1
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Andrew SukMIT
1
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Disjoint edges in complete topological graphs
Definitions
v0
1
0
0
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vl
1
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vi
vj
1
0
0
1
vk
1
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0
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Assume edges cross.
Andrew SukMIT
Disjoint edges in complete topological graphs
Definitions
v0
1
0
0
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vl
1
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vi
vj
1
0
0
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vk
1
0
0
1
1
0
0
1
′ stabs v v , which is a contradiction.
Sk,l
i j
Andrew SukMIT
Disjoint edges in complete topological graphs
Definitions
v0
1
0
1
0
vl
1
0
0
1
vi
vj
1
0
0
1
vk
1
0
0
1
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0
1
0
Two edges must be disjoint.
Andrew SukMIT
Disjoint edges in complete topological graphs
Definitions
Same argument shows
v0
1
0
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vk
v1
l
0
vi
1
0
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Andrew SukMIT
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vj
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Disjoint edges in complete topological graphs
Definitions
Ω(n1/3 ) pairwise disjoint edges in Kn+1 .
v011
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Andrew SukMIT
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Disjoint edges in complete topological graphs
Definitions
Open Problems.
1
Can the Ω(n1/3 ) bound be improved? Perhaps to Ω(n1/2 )?
2
∗ (m) = O(m2 ).
Just need to show πF
3
Best known upper bound construction: O(n) pairwise disjoint
edges.
Andrew SukMIT
Disjoint edges in complete topological graphs
Definitions
Pairwise crossing edges in Kn ?
Theorem (Fox and Pach, 2008)
Every complete n-vertex simple topological graph has at least
Ω(nδ ) pairwise crossing edges.
δ ≈ 1/50.
Problem: Can one improve this bound?
Andrew SukMIT
Disjoint edges in complete topological graphs
Definitions
This problem is interesting for complete geometric graphs (edges
are straight line segments)!
Theorem (Aronov, Erdős, Goddard, Kleitman, Klugerman, Pach,
Schulman, 1997)
Every complete n-vertex geometric graph has at least Ω(n1/2 )
pairwise crossing edges.
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Andrew SukMIT
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Disjoint edges in complete topological graphs
Definitions
This problem is interesting for complete geometric graphs!
Theorem (Aronov, Erdős, Goddard, Kleitman, Klugerman, Pach,
Schulman, 1997)
Every complete n-vertex geometric graph has at least Ω(n1/2 )
pairwise crossing edges.
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Conjecture: Ω(n).
Andrew SukMIT
Disjoint edges in complete topological graphs
Definitions
This problem is interesting for complete geometric graphs!
Theorem (Aronov, Erdős, Goddard, Kleitman, Klugerman, Pach,
Schulman, 1997)
Every complete n-vertex geometric graph has at least Ω(n1/2 )
pairwise crossing edges.
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Andrew SukMIT
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Disjoint edges in complete topological graphs
Definitions
Similar dual problems.
Theorem
Every n-vertex topological graph with no crossing edges contains
at most 3n − 6 = O(n) edges.
Relaxation of planarity.
Conjecture
Every n-vertex topological graph with no k pairwise crossing edges
contains at most O(n) edges.
Andrew SukMIT
Disjoint edges in complete topological graphs
Definitions
Special Cases
1
Conjecture is true for k = 3, 4 (k = 3 by Agarwal et. al. ’97,
Pach, Radoičić, Tóth ’03, Ackerman and Tardos ’08, k = 4 by
Ackerman ’09)
Best known bound for general k:
1
Every n-vertex (not simple) topological graph with no k
pairwise crossing edges has at most n(log n)O(log k) edges.
(Fox and Pach, 2008)
2
Every n-vertex simple topological graph with no k pairwise
c
crossing edges has at most (n log n) · 2α k (n) edges. (Fox,
Pach, Suk, 2011)
Andrew SukMIT
Disjoint edges in complete topological graphs
Definitions
Problem (Erdős)
Let S be a family of n segments in the plane, such that no three
members pairwise cross. Can you color the members in S with at
most c colors, such that each color class consists of pairwise
disjoint segments.
Best known O(log n), McGuinness 1997.
Andrew SukMIT
Disjoint edges in complete topological graphs
Definitions
Problem (Erdős)
Let S be a family of n segments in the plane, such that no three
members pairwise cross. Can you color the members in S with at
most c colors, such that each color class consists of pairwise
disjoint segments.
Best known O(log n), McGuinness 1997.
Is there a subcollection of Ω(n) pairwise disjoint segments?
Andrew SukMIT
Disjoint edges in complete topological graphs
Definitions
Problem (Erdős)
Let S be a family of n segments in the plane, such that no three
members pairwise cross. Can you color the members in S with at
most c colors, such that each color class consists of pairwise
disjoint segments.
Best known O(log n), McGuinness 1997.
Is there a subcollection of Ω(n) pairwise disjoint segments? Best
known Ω(n/ log n).
Andrew SukMIT
Disjoint edges in complete topological graphs
Definitions
Thank you!
Andrew SukMIT
Disjoint edges in complete topological graphs