Chromatic Ramsey Number andCircular Chromatic Ramsey Number

Xuding Zhu

Department of MathematicsZhejiang Normal University

Among 6 people,

There are 3 know each other, or 3 do not know each other.

Know each other

Do not know each other

Among 6 people,

There are 3 know each other, or 3 do not know each other.

Among 6 people,

There are 3 know each other, or 3 do not know each other.

Among 6 people,

There are 3 know each other, or 3 do not know each other.

Among 6 people,

There are 3 know each other, or 3 do not know each other.

Colour the edges of by red or blue,there is either a red or a blue

6K3K 3K

Theorem [Ramsey] For any graphs G and H, there exists a graph F such that if the edges of F are coloured by red and blue, then there is a red copy of G or a blue copy of H

For `any’ systems , there exists a system F such that if `elements’ of F are partitioned into k parts, then for some i, the ith part contains as a subsystem.

k)i (1 i G

iG

General Ramsey Type Theorem:

Sufficiently large or complicated

“Complete disorder is impossible”

A sufficiently large scale (or complicated) system must contains an interesting sub-system.

There are Ramsey type theorems in many branches of mathematics such as combinatorics, number theory, geometry, ergodic theory, topology, combinatorial geometry, set theory, and measure theory.

Ramsey Theory has a wide range of applications.

If the k-tuples M are t-colored, then

Theorem [Ramsey, 1927]

,,, knt,|| with set aFor mMM

nMMM |'| ,'

all the k-tuples of M’ having the same color.

principle pigeonhole :1k

graph complete a of edges thecolouring :2k

m

For any partition of integers into finitely many parts, one part contains arithematical progression of arbitrarylarge length.

Van der Waeden Theorem

Szemerédi's theorem (1975)

Every set of integers A with positive density contains

arithematical progression of arbitrary length.

Timonthy Gowers [2001] gave a proof using both Fourier analysis and combinatorics.

Regularity lemmaErdos and Turan conjecture (1936)

Harmonic analysis

Ramsey number R(3,k)

For any 2-colouring of the edges of F with colours red and blue,there is a red copy of G or a blue copy of H.

),(K 336 KK

),( HGF means.

),(K 335 KK

),( :min HGKnR(G,H) n

),(),( lkRKKR lk

The Ramsey number of (G,H) is

),( :min ,3 3 kn KKKnk)R(

1933, George Szekeres, Esther Klein, Paul Erdos

starting with a geometric problem, Szekeres re-discovered Ramsey theorem, and proved 2 ,3 kk)R(

Erdos [1946]

3/2

lnk

k ,3 k)R(

Erdos [1961]

Graver-Yackel [1968]

k

kOk)R(

ln

lnln k ,3 2

2

lnk

k ,3 k)R(

Ajtai-Komlos-Szemeredi[1980]

k

kOk)R(

ln ,3

2

Kim [1995]

k

kk)R(

ln ,3

2

2 ,3 kk)R( Szekere [1933]

Many sophisticated probabilistic tools are developed

George Szekere and Esther Klein marriedlived together for 70 year, died on the same day 2005.8.28, within one hour.

),( :min lkn KKKnR(k,l)

Bounds for R(k,l)

k l 3 4 5 6 7 8

3 6 9 14 18 23 28

4 18 25 36

41

49

61

58

84

5 43

49

58

87

80

143

101

216

6 102

165

113

298

132

495

7 205

540

217

1031

8 282

1870

Bounds for R(k,l)

k l 3 4 5 6 7 8

3 6 9 14 18 23 28

4 18 25 36

41

49

61

58

84

5 43

49

58

87

80

143

101

216

6 102

165

113

298

132

495

7 205

540

217

1031

8 282

1870

Bounds for R(k,l)

k l 3 4 5 6 7 8

3 6 9 14 18 23 28

4 18 25 36

41

49

61

58

84

5 43

49

58

87

80

143

101

216

6 102

165

113

298

132

495

7 205

540

217

1031

8 282

1870

How to measure a system?

A sufficiently large scale (or complicated) system must contains an interesting sub-system.

What is large scale?

What is complicated?

How to measure a graph?

),( : min

is ,( ofnumber Ramsey The

HGKnR(G,H)

H)G

n

),( |:)(| min

is ,( ofnumber Ramsey The

HGFFVR(G,H)

H)G

),( |:)(| min

is ,( ofnumber Ramsey Size The

HGFFE(G,H)R

H)G

E

),( |:)(| min

is ,( ofnumber Ramsey -degree-max The

HGFF(G,H)R

H)G

),( :)( min

is ,( ofnumber Ramsey chromatic The

HGFF(G,H)R

H)G

),( :)( inf

is ,( ofnumber Ramsey chromaticcircular The

HGFF(G,H)R

H)G

cc

Chromatic number

Circular chromatic number

1,...,1,0: kVf

)()(~ yfxfyx

G=(V,E): a graph

an integer:1k

3k

An k-colouring of G is

0

1

20

1

A 3-colouring of 5Csuch that

The chromatic number of G is

colouring-k a has G :kmin )( G

Vf :

)()( yfxf

G=(V,E): a graph

an integer:1k

k-colouring of G is

such that yx ~

An

1|)()(|1 kyfxf

a real number

A (circular)

1,...,1,0 k),0[ k

0

1

20.5

1.5

A 2.5-coloring

1r

r-colouring of G is

),0[ r

1|)()(|1 ryfxf

The circular chromatic number of G is

)(Gc { r: G has a circular r-colouring }infmin

f is k-colouring of G

Therefore for any graph G,

)()( GGc

f is a circular k-colouring of G

0=r

3

1

24

x~y |f(x)-f(y)|_r ≥ 1

The distance between p, p’ in the circle is

|'| |,'| min|p'-p| r pprpp

f is a circular r-colouring if

0 r

pp’

Basic relation between )(G and )(Gc

).()(1)( GGG c

Circular chromatic number of a graph is a refinementof its chromatic number.

Graph coloring is a model for resource distribution

Circular graph coloring is a model for resource distributionof periodic nature.

),( :)( min

is ,( ofnumber Ramsey chromatic The

HGFF(G,H)R

H)G

Introduced by Burr-Erdos-Lovasz in 1976

),( lkR),K(KR lk

),( , HGRH)(GR

),( GGR(G)R

If F has chromatic number , then there is a2 edge colouring of F in which each monochromaticsubgraph has chromatic number n-1.

2)1( n

),( GGF for any n-chromatic G.

4n

If F has chromatic number , then there is a2 edge colouring of F in which each monochromaticsubgraph has chromatic number n-1.

2)1( n

),( GGF for any n-chromatic G.

1)1()( then ,)( If :nObservatio 2 nGRnG

1)1()( and )(G with graph a is there

n,each For :1976] Lovasz,-Erdos-[Burr Conjecture2 nGRnG

Could be much larger

1)1()( and )(G with graph a is there

n,each For :1976] Lovasz,-Erdos-[Burr Conjecture2 nGRnG

The conjecture is true for n=3,4 (Burr-Erdos-Lovasz, 1976)

The conjecture is true for n=5 (Zhu, 1992)

The conjecture is true (Zhu, 2011)

Attempts by Tardif, West, etc. on non-diagonal casesof chromatic Ramsey numbers of graphs.

There are some upper bounds on nGG )(:)(Rmin

No more other case of the conjecture were verified, until 2011

1)1()(:)(Rmin 2 nnGG

)hom(Kn

Lovasz]-Erdos-[Burr Lemma

n G(G)R

For any 2 edge-colouring of Kn, there is a monochromaticgraph which is a homomorphic image of G.

Graph homomorphism = edge preserving map

GH

1)1()( and )(G with graph a is there

n,each For :1976] Lovasz,-Erdos-[Burr Conjecture2 nGRnG

To prove Burr-Erdos-Lovasz conjecture for n, we need to construct an n-chromatic graph G, so that any 2 edge colouring of has a monochromatic subgraph which is a homomorphic image of G.

1)1( 2nK

The construction of G is easy:

Take all 2 edge colourings of 1)1( 2nK

mccc ,,, 21

For each 2 edge colouring ci of , one of the monochromaticsubgraph, say Gi, , has chromatic number at least n.

1)1( 2nK

iG

To prove this conjecture for n, we need to construct an n-chromatic graph G, so that any 2 edge colouring of has a monochromatic subgraph which is a homomorphic image of G.

1)1( 2nK

The construction of G is easy:

Take all 2 edge colourings of 1)1( 2nK

mccc ,,, 21

For each 2 edge colouring of , one of the monochromaticsubgraph, say Gi, , has chromatic number at least n.

1)1( 2nK

iGic

mGGGG 21

HG

G

H

GxH

Projections are homomorphisms

To prove this conjecture for n, we need to construct an n-chromatic graph G, so that any 2 edge colouring of has a monochromatic subgraph which is a homomorphic image of G.

1)1( 2nK

The construction of G is easy:

Take all 2 edge colourings of 1)1( 2nK

mccc ,,, 21

For each 2 edge colouring ci of , one of the monochromaticsubgraph, say Gi, , has chromatic number at least n.

1)1( 2nK

mGGGG 21

GGi of image chomomorphi a is Each

iG

?

G

H

)(),(min)(

:1966] i,[Hedetniem Conjecture

HGHG

To prove this conjecture for n, we need to construct an n-chromatic graph G, so that any 2 edge colouring of has a monochromatic subgraph which is a homomorphic image of G.

1)1( 2nK

?

If Hedetniemi’s conjecture is true, then

Burr-Erdos-Lovasz conjecture is true.

GG of setst independen offamily :)(

A k-colouring of G partition V(G) into k independent sets.

1,0)(: G

1)()(,

GXXv

X

)(

)(minGX

X)(G

integer linear programming

GG of setst independen offamily :)(

A k-colouring of G partition V(G) into k independent sets.

1,0)(: G

1)()(,

GXXv

X

)(

)(minGX

X)(G

]1,0[

)(Gf

ofnumber chromatic fractional :)( GGf

linear programming

)()( GGf

)(),(min)(

:1966] i,[Hedetniem Conjecture

HGHG

)(),(min)(

:2002] [Z, Conjecture

HGHG fff

Fractional Hedetniemi’s conjecture

To prove this conjecture for n, we need to construct an n-chromatic graph G, so that any 2 edge colouring of has a monochromatic subgraph which is a homomorphic image of G.

1)1( 2nK

If Hedetniemi’s conjecture is true, then

Burr-Erdos-Lovasz conjecture is true.

Observation: If fractional Hedetniemi’s conjecture is true, then Burr-Erdos-Lovasz conjecture is true.

To prove this conjecture for n, we need to construct an n-chromatic graph G, so that any 2 edge colouring of has a monochromatic subgraph which is a homomorphic image of G.

1)1( 2nK

The construction of G is easy:

Take all 2 edge colourings of 1)1( 2nK

mccc ,,, 21

For each 2 edge colouring ci of , one of the monochromaticsubgraph, say Gi, , has chromatic number at least n.

1)1( 2nK

mGGGG 21

GGi of image chomomorphi a is Each

iG fractional chromatic number > n-1

1)( nGf 1)()( nGG f

)(),(min)(

:1966] i,[Hedetniem Conjecture

HGHG

)(),(min)(

:2002] [Z, Conjecture

HGHG fff

Fractional Hedetniemi’s conjecture

)(),(min)(

:2002] [Z, Conjecture

HGHG fff

Theorem [Huajun Zhang, 2011]

If both G and H are vertex transitive, then

Theorem [Z, 2011]

GG of setst independen offamily :)(

A k-colouring of G partition V(G) into k independent sets.

1,0)(: G

1)()(,

GXXv

X

)(

)(minGX

X)(G

]1,0[

)(Gf

ofnumber chromatic fractional :)( GGf

linear programming

]1,0[)(: GV

dual problem

1)()(,

GXXv

v

)(

)(maxGVv

v)(Gf

ofnumber clique fractional :)( GGf

The fractional chromatic number of G is obtained by solving a linear programming problem

The fractional clique number of G is obtained by solving its dual problem

)()( GG ff

)(),(min)(

:2002] [Z, Conjecture

HGHG fff

Fractional Hedetniemi’s conjecture is true

Theorem [Z, 2010]

)(),(min)( HGHG fff

:sketch oofPr

)(),(min weight with total

of clique fractional aconstruct tosuffices

HG

HG

ff

)(),(min)( HGHG fff

)(),(min)( HGHG fff

)(),(min)( HGHG fff

Easy!

Difficult!

)(),(min weight with total

of clique fractional aconstruct tosuffices

HG

HG

ff

Easy

Easy

Difficult

G of clique fractional maximum a ,10 : ],[V(G)g

H of clique fractional maximum a ],1,0[)( : HVh

(H)(G),ωω

g(x)h(y)(x,y)

HGV

ffmax

as defined ],1,0[)( :

)(),(min weight with total

of clique fractional a is

HG

HG

ff

Uyxff HGyhxg

HGU

),(

)(),(max)()(

, of set t independen

Easy!

Difficult!

),( :)( inf

is ,( ofnumber Ramsey chromaticcircular The

HGFF(G,H)R

H)G

cc

G)(GR(G)Rcc

,

What is the relation between and ? (G)Rc c(G)

Basic relation between )(G and )(Gc

).()(1)( GGG c

).()( GGc

)(Gc )(Gis a refinement of

)(G )(Gcis an approximation of

There are many periodical scheduling problems in computer sciences.

The reciprocal of is studied by computer scientists as efficiency of a certain scheduling method, in 1986.

).(Gc

Circular colouring is a good model for periodical scheduling problems

zGGRzR ccc )(:)(inf)(

nGGRnR )(:)(min)(

1)1()( :1976] Lovasz,-Erdos-[Burr Conjecture 2 nnR Theorem [Zhu, 2011]

?)( zRc No conjecture yet!

).()()( GGG cf

.1)()( GG c

.largey arbitraril becan )()( GG f

)1( )( then ,integeran is 2 If kkkRkc

Using fractional version of Hedetniemi’s conjecture,Jao-Tardif-West-Zhu proved in 2014

)12(2 )2( c

R )13(3 )( 3 KRc

?)13(3 )3( c

R

2 rational real becan )( Gc

?)( of valuespossible theareWhat x

GR

),( :)( inf GGFF(G)R cc

colouring-circular a has : inf rGr(G)c min

min ?

No !

[ Jao-Tardif-West-Zhu, 2014]

4 )( then ,2

52 If zRz

c

6 )( 3 KRc

9 ),( 43 KKRc

5 ),(3

14 53 CCR

c

2

9 ),(4 73 CCR

c

Some other results by Jao-Tardif-West-Zhu, 2014

謝謝