Which graphs are extremal?

Which graphs are extremal?

László LovászEötvös Loránd University

Budapest

September 2012 1

September 2012

Turán’s Theorem (special case proved by Mantel):G contains no triangles #edgesn2/4

Theorem (Goodman):

3#edges #triangles (2 -1) ( )2 3n n

c c c o n

Extremal:

2

Some old and new results from extremal graph theory

September 2012 3


| ( )|

hom( , )| ( ) |

( , ) V F

F GV G

t F G Probability that random map

V(F)V(G) preserves edges

Homomorphism: adjacency-preserving map

September 2012 4


Theorem (Goodman):

3#edges #triangles (2 -1) ( )2 3n n

c c c o n

t( ,G) – 2t( ,G) + t( ,G) ≥ 0

t( ,G) = t( ,G)2

September 2012

Kruskal-Katona Theorem (very special case):

#edges #triangles 2 3k k

nk

5


t( ,G)2 ≥ t( ,G)3

t( ,G) ≥ t( ,G)

September 2012

Semidefiniteness and extremal graph theory Tricky examples

1

10

Kruskal-Katona

Bollobás

1/2 2/3 3/4

Razborov 2006

Mantel-Turán

Goodman

Fisher

Lovász-Simonovits


6

September 2012

Theorem (Erdős):G contains no 4-cycles #edgesn3/2/2

(Extremal: conjugacy graph of finite projective planes)

( )4 4#edges #4-cycles 2 4n n

c c o n

7


t( ,G) ≥ t( ,G)4

September 2012 8

General questions about extremal graphs

- Is there always an extremal graph?

- Which inequalities between subgraph densities are valid?

- Which graphs are extremal?

- Can all valid inequalities be proved using just Cauchy-Schwarz?

September 2012 9






September 2012 10

Which inequalities between densities are valid?

1

( , ): 0? ?m

ii iG a t F G

If valid for large G,

then valid for all

April-May 2013 11

Analogy with polynomials

p(x1,...,xn)0

for all x1,...,xnZ undecidable Matiyasevich

for all x1,...,xnR decidable Tarski

Û p = r12 + ...+ rm

2 (r1, ...,rm: rational functions)

„Positivstellensatz” Artin

September 2012 12


Undecidable…

Hatami-Norine

1

( , ): 0? ?m

ii iG a t F G

September 2012

1

10 1/2 2/3 3/4

13

The main trick in the proof

t( ,G) – 2t( ,G) + t( ,G) = 0 …

September 2012 14


Undecidable…

Hatami-Norine

…but decidable with an arbitrarily small error.

L-Szegedy

September 2012 15






September 2012

Write a ≥ 0 if t(a,G) ≥ 0 for every graph G.

Goodman:

Computing with graphs

16

-2 + 0

Kruskal-Katona: - 0

Erdős: - 0

1

1

"quantum graph

( , )

"

( , )

m

i

m

i

i i

i i

a F

a t F G

a

t a G

September 2012

- +-2

= - +-

- +- 2+2 2- = - +- +2 -4 +2


17

+- 2

2- = -2 +

Goodman’s Theorem-2 + 0

September 2012

2 221 1

2( ... . .) ?. mn xz y yz

Question: Suppose that x ≥ 0. Does it follow that2 21 .. ?. mx y y

Positivstellensatz for graphs?

18

No! Hatami-Norine

If a quantum graph x is sum of squares (ignoring labels and isolated nodes), then x ≥ 0.

September 2012

A weak Positivstellensatz

19

L - Szegedy

2

,1 1

0 i j ji j

x x F

(ignoring labels and isolated nodes)

September 2012 20






Minimize over x03 6x x-

minimum is not attainedin rationals

Minimize t(C4,G) over graphs with edge-density 1/2

minimum is not attainedamong graphs

always >1/16,arbitrarily close for random

graphs

Real numbers are useful

Graph limits are useful

September 2012 21

Is there always an extremal graph?

Quasirandom

graphs

September 2012

20 : [0,1] [0,1] symmetric, measurableW W

Limit objects

22

(graphons)

G

0 0 1 0 0 1 1 0 0 0 1 0 0 10 0 1 0 1 0 1 0 0 0 0 0 1 01 1 0 1 0 1 1 1 1 0 1 0 1 10 0 1 0 1 0 1 0 1 0 1 1 0 00 1 0 1 0 1 1 0 0 0 1 0 0 11 0 1 0 1 0 1 1 0 1 1 1 0 11 1 1 1 1 1 0 1 0 1 1 1 1 00 0 1 0 0 1 1 0 1 0 1 0 1 10 0 1 1 0 0 0 1 1 1 0 1 0 00 0 0 0 0 1 1 0 1 0 1 0 1 01 0 1 1 1 1 1 1 0 1 0 1 1 10 0 0 1 0 1 1 0 1 0 1 0 1 00 1 1 0 0 0 1 1 0 1 1 1 0 11 0 1 0 1 1 0 1 0 0 1 0 1 0 AG

WG

Graphs Graphons

September 2012 23

September 2012

( ) ( )[0,1]

( , )( , )V F

i jij E F

W x x dxt F W

Limit objects

24

(graphons)

( , ( ): ) ,nn F t F G WW tG F

20 : [0,1] [0,1] symmetric, measurableW W

t(F,WG)=t(F,G)

(G1,G2,…) convergent: F t(F,Gn) converges

Borgs-Chayes-L-Sós-Vesztergombi

A random graph with 100 nodes and with 2500 edges

April-May 2013 25

Example: graph limit

1 max( , )- x y

April-May 2013 26

A randomly grown uniform attachment graph on 200 nodes

Example: graph limit

April-May 2013 27

Limit objects: the math

For every convergent graph sequence (Gn)

there is a WW0 such that GnW .

Conversely, W (Gn) such that GnW .

L-Szegedy

W is essentially unique

(up to measure-preserving

transformation).Borgs-Chayes-L

September 2012

k=2:

...

...

( )f

M(f, k)

28

Connection matrices

September 2012

W: f = t(.,W)

k M(f,k) is positive semidefinite,

f()=1 and f is multiplicative

Semidefinite connection matrices

29

f: graph parameter

L-Szegedy

the optimum of a semidefinite program is 0:

minimize

subject to M(x,k) positive semidefinite k

x(K0)=1

x(GK1)=x(G)

September 2012

Proof of the weak Positivstellensatz (sketch2)

Apply Duality Theorem of semidefinite programming

30

0: ( , )i iG a t F G

( )i ia x F

September 2012 31

Is there always an extremal graph?

No, but there is always an extremal graphon.

The space of graphonsis compact.

September 2012 32






Given quantum graphs g0,g1,…,gm,

find max t(g0,W)

subject to t(g1,W) = 0

…

t(gm,W) = 0

September 2012 33

Extremal graphon problem

Finite forcing

Graphon W is finitely forcible:

1 1

1

( , ) ( , )( , ) ( , )

( , ) ( , )

,..., m

m m

t F U t F WF t F U

F F U

t F Wt F U t F W

M

Every finitely forcible graphon is extremal:

minimize 21 1

1

( ( , ) )m

jt F U

Every unique extremal graphon is finitely forcible.?? Every extremal graph problem has a finitely forcible extremal graphon ??

September 2012 34

Finitely forcible graphons

2

3

2( , )32( , )9

t K W

t K W

Goodman

1/22

4

1( , )21( , )

16

t K W

t C W

Graham-Chung-Wilson

September 2012 35


Stepfunctions finite graphs with node and edgeweights

Stepfunction:

September 2012 36

Which graphs are extremal?

Stepfunctions are finitely forcible L – V.T.Sós

,

( , )

( , ) ( , )2 1 2

0

16

t W

t K W t K W

September 2012 37


Is the following graphon finitely forcible?

angle <π/2

September 2012 38

Which graphons are finitely forcible?

April-May 2013 39

Thanks, that’sall for today!

September 2012 40

The Simonovits-Sidorenko Conjecture

F bipartite, G arbitrary t(F,G) ≥ t(K2,G)|E(F)|

Known when F is a tree, cycle, complete bipartite… Sidorenko

F is hypercube HatamiF has a node connected to all nodesin the other color class Conlon,Fox,Sudakov

F is "composable" Li, Szegedy

?

September 2012 41

The Simonovits-Sidorenko Conjecture

Two extremal problems in one:

For fixed G and |E(F)|, t(F,G) is minimized

by F= …

asymptotically

For fixed F and t( ,G), t(F,G) is minimized

by random G

September 2012 42

The integral version

Let WW0, W≥0, ∫W=1. Let F be bipartite. Then t(F,W)≥1.

For fixed F, t(F,W) is minimized over W≥0, ∫W=1

by W1

?

September 2012 43

The local version

Let 1 11 1 , 1

4 | ( ) | 4 | ( ) |W W

E F E F

Then t(F,W) 1.

September 2012 44

The idea of the proof

'

( , ) ( ,1 ) ( ', )F F

t F W t F U t F U

( , ) 1( , )

( , ) ( , )

...( , ) ...

t F Wt U

t U t U

t U

00<

September 2012 45

The idea of the proof

Main Lemma:

If -1≤ U ≤ 1, shortest cycle in F is C2r,

then t(F,U) ≤ t(C2r,U).

September 2012 46

Common graphs

1 14 2( , ) ( , ) (1) 2 , ,( )( )t G t G o t G n V V V

1 14 2( , ) ( ,1 ) 2 ,( )t W t W t V V V

4 4 41 1

32 2( , ) ( ,1 ) 2 ,( )t K W t K W t K Erdős: ?

Thomason

September 2012 47

Common graphs

F common:

12( , ) ( ,1 ) 2 ,( )t W t W t

Hatami, Hladky, Kral, Norine, Razborov

12( , ) ( ,1 ) 2 ,( )t F W t F W t F W

Common graphs:

Sidorenko graphs (bipartite?)

Non-common graphs:

graph containingJagger, Stovícek, Thomason

Common graphs

1 1 1, , 2 ,2 2 2U Ut F t F t F

,1 ,1 2t F U t F U

September 2012 48

12( , ) ( ,1 ) 2 ,( )t F W t F W t F

'

,1 ( ', )F F

t F U t F U

'

( ) 0 (2)

,1 ( ,1 ) ( ', )F F

E F

t F U t F U t F U

Common graphs

September 2012 49

'( ) 0 (2)

1 1 ( ', ) 0F F

E F

U t F U

F common:

8 , 2 , , 4 ,t U t U t U t U

is common. Franek-Rödl

8 +2 + +4

= 4 +2 +( +2 )2 +4( - )

Common graphs

F locally common:

12

12( , ) ( ,1 ) 2 "c, lose to "( )t F W t F W t F W

1 1 0 ,1 ,1 2U t F U t F U

September 2012 50

12 +3 +3 +12 +

12 2 +3 2 +3 4 +12 4 + 6

is locally common. Franek-Rödl

Common graphs

September 2012 51

graph containing is locally common.

graph containing is locally common

but not common.

Not locally common:

Common graphs

September 2012 52

'( ) 0 (2)

1 1 ( ', ) 0F F

E F

U t F U

F common:

8 , 2 , , 4 ,t U t U t U t U

- 1/2 1/2 - 1/2 1/2

8 +2 + +4 = 4 +2 +( -2 )2

is common. Franek-Rödl

September 2012 53

Common graphs

F common:

12( , ) ( ,1 ) 2 ,( )t W t W t

Hatami, Hladky, Kral, Norine, Razborov

12( , ) ( ,1 ) 2 ,( )t F W t F W t F W

Common graphs:

Sidorenko graphs (bipartite?)

Non-common graphs:

graph containingJagger, Stovícek, Thomason

September 2012

Theorem (Erdős-Stone-Simonovits): (F)=3

54


{ }2

max ( ) : ( ) ,4nE G V G n F G := Ë

22

2/2, /2 /2, /2

If and ( ) ( ), then there is a4

on ( ) suchthat ( ) ( ) ( ).n n n n

nF G E G o n

K V G E G E K o nV

Ë ³ -

=

September 2012

Graph parameter: isomorphism-invariant function on finite graphs

k-labeled graph: k nodes labeled 1,...,k, any number of unlabeled nodes

1

2

55


1

0

( , )W x y dx d y d-regular graphon:

2

22,1

( , )

( , )

t K W d

t K W d

d-regular

( , ) 0t W

September 2012 56

Finitely expressible properties

( , ) 0t W W is 0-1 valued, and can be rearrangedto be monotone decreasing in both variables

"W is 0-1 valued" is not finitely expressible in terms of simple gaphs.

( , ) ( , )t W t W W is 0-1 valued

September 2012 57

Finitely expressible properties

Which graphs are extremal?

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