Extremal graph theory and limits of graphs László Lovász September 20121

Extremal graph theory

and limits of graphs

László Lovász

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 3

n nc c c o n

Extremal:

2

Some old and new results from extremal graph theory

September 2012

Kruskal-Katona Theorem (very special case):

#edges #triangles 2 3

k k

nk

3


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


4

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 4

n nc c o n

5


September 2012

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

6


{ }2

max ( ) : ( ) ,4

nE 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 7

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 8






September 2012

: # of homomorphismsho ofm( , ) intoG F GF

| ( )|

hom( , )

| ( ) |( , )

V F

F G

V Gt F G Probability that random map

V(F)V(G) is a hom

9

Homomorphism functions

Homomorphism: adjacency-preserving map

1

( , ): 0? ?m

ii iG t F G

If valid for large G,

then valid for all

September 2012 10






September 2012 11

Which inequalities between densities are valid?

Undecidable…

Hatami-Norine

September 2012

1

10 1/2 2/3 3/4

12

The main trick in the proof

t( ,G) – 2t( ,G) + t( ,G) = 0 …

September 2012 13

Which inequalities between densities are valid?

Undecidable…

Hatami-Norine

…but decidable with an arbitrarily small error.

L-Szegedy

September 2012 14






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

15

Which parameters are homomorphism functions?

September 2012

k=2:

...

...

( )f

M(f, k)

16

Connection matrices

September 2012

f = hom(.,H) for some weighted graph H

M(f,k) is positive semidefinite

and has rank ck

Freedman - L - Schrijver

Which parameters are homomorphism functions?

17

September 2012

k-labeled quantum graph:

GG

x x G finite formal sum ofk-labeled graphs

1

2

infinite dimensional linear space

18

Computing with graphs

Gk = {k-labeled quantum graphs}

September 2012

kG is a commutative algebra with unit element ...

Define products:

1 2

1 2

1 2,

G G GG GG

GG

x y Gx G GG yæ öæ ö÷ ÷ç ç÷ ÷ç ç÷ ÷ç ç÷ ÷è øè ø

=å å å

19


G1,G2: k-labeled graphsG1G2 = G1G2, labeled nodes identified

September 2012

Inner product:

(, )' 'f

fG GGG

f: graph parameter

, ,f f

x yz xy z

( )GGG G

x Gf x G f

:,f

x y extend linearly

20


September 2012

2

0

0

,

( )

( , ) is positive semidefinite

kf

k

x x x

f x x

M f k

G

G

f is reflection positive


21

September 2012

Write x ≥ 0 if hom(x,G) ≥ 0 for every graph G.

Turán: -2 + 0³

Kruskal-Katona: - 0³

Blakley-Roy: - 0³


22

September 2012

- +-2

= - +-

- +- 2+2

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

Goodman’s Theorem


23

+- 2+- 2 ≥ 0

2- = 2 -4 +2

t( ,G) – 2t( ,G) + t( ,G) ≥ 0

September 2012

2 221 1

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

Question: Suppose that x ≥ 0. Does it follow that

2 21 .. ?. mx y y

Positivstellensatz for graphs?

24

No! Hatami-Norine

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

September 2012

Let x be a quantum graph. Then x 0

2 2

1 1 10 ,..., ...m k mk y y G x y y

A weak Positivstellensatz

25

L-Szegedy

the optimum of a semidefinite program is 0:

minimize

subject to M(f,k) positive semidefinite for all k

f(K1)=1

f(GK1)=f(G)

September 2012

Proof of the weak Positivstellensatz (sketch2)

Apply Duality Theorem of semidefinite programming

26

0: ( , )i iG t F G

( )i if F

September 2012 27






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 28

Is there always an extremal graph?

Quasirandom graphs

September 2012

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

Limit objects

29

(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 30

September 2012

( ) ( )[0,1]

( , )( , )V F

i jij E F

W x x dxt F W

Limit objects

31

(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

For every convergent graph sequence (Gn)

there is a graphon W such that GnW.

September 2012 32

Limit objects

LS

For every graphon W there is a graph

sequence (Gn) such that GnW. LS

W is essentially unique (up to measure-preserving transformation). BCL

September 2012 33

Is there always an extremal graph?

No, but there is always an extremal graphon.

The space of graphonsis compact.

September 2012

f = t(.,W)

k M(f,k) is positive semidefinite,

f()=1 and f is multiplicative

Semidefinite connection matrices

34

f: graph parameter

September 2012 35






Given quantum graphs g0,g1,…,gm,

find max t(g0,W)

subject to t(g1,W) = 0

…

t(gm,W) = 0

September 2012 36

Extremal graphon problem

Finite forcing

Graphon W is finitely forcible: 1

1

1

1( ,

,..., , ,...,

)

( , ) ( , )

( , )

:m m

m m

t F U

F t F U t F W

F

t F U

F

M

Every finitely forcible graphon is extremal:

minimize 21 1

1

( ( , ) )m

j

t F U

Every unique extremal graphon is finitely forcible.

?? Every extremal graph problem has a finitely forcible extremal graphon ??

September 2012 37

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 38


Stepfunctions finite graphs with node and edgeweights

Stepfunction:

September 2012 39

Which graphs are extremal?

Stepfunctions are finitely forcible L – V.T.Sós

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 40

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 41

Finitely expressible properties

,

( , )

( , ) ( , )2 1 2

0

1

6

t W

t K W t K W

p(x,y)=0 p monotone decreasingsymmetric polynomial

finitely forcible

?

September 2012 42


( , ) 0t W = Þ Sp(x,y)=0

1 1 1, 1( , ) ( , )a b a b

a b

S S

t K W ab x y dx dy b x y n x y e dx

21 2 ,( , ) ( , ) ( ) ( , )

ii

iS

a bx y p x y n x y e e ds t K W

a

Stokes 1 2

1, , 1

( , ) ( )

( 1) ( , ) ( 1) ( , )

a b

a b a b

S

x y n x y e e ds

a t K W b t K W

September 2012 43


Is the following graphon finitely forcible?

angle <π/2

September 2012 44


September 2012 45

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 46

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 47

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 48

The local version

Let 1 1

1 1 , 14 | ( ) | 4 | ( ) |

W WE F E F

Then t(F,W) 1.

September 2012 49

The idea of the proof

'

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

t F W t F U t F U

( , ) 1

( , )

( , ) ( , )

...

( , ) ...

t F W

t U

t U t U

t U

00<

September 2012 50

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 51

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 52

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 2

U Ut F t F t F

,1 ,1 2t F U t F U

September 2012 53

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 54

'( ) 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 55

12 +3 +3 +12 +

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

is locally common. Franek-Rödl

Common graphs

September 2012 56

graph containing is locally common.

graph containing is locally common

but not common.

Not locally common:

Common graphs

September 2012 57

'( ) 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 58

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

Documents

Extremal graph theory and limits of graphs László Lovász September 20121