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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
Some old and new results from extremal graph theory
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
Some old and new results from extremal graph theory
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
Some old and new results from extremal graph theory
September 2012
Theorem (Erdős-Stone-Simonovits): (F)=3
6
Some old and new results from extremal graph theory
{ }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
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
: # 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
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 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
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
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
Computing with graphs
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
Computing with graphs
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
Computing with graphs
21
September 2012
Write x ≥ 0 if hom(x,G) ≥ 0 for every graph G.
Turán: -2 + 0³
Kruskal-Katona: - 0³
Blakley-Roy: - 0³
Computing with graphs
22
September 2012
- +-2
= - +-
- +- 2+2
2- = - +- +2 -4 +2
Goodman’s Theorem
Computing with graphs
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
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?
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
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?
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
Finitely forcible graphons
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
Finitely forcible graphons
( , ) 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
Finitely forcible graphons
Is the following graphon finitely forcible?
angle <π/2
September 2012 44
Finitely forcible graphons
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