Chromatic Ramsey Number and
Download
1 / 65

Chromatic Ramsey Number and Circular Chromatic Ramsey Number - PowerPoint PPT Presentation


  • 157 Views
  • Uploaded on

Chromatic Ramsey Number and Circular Chromatic Ramsey Number. Xuding Zhu. Department of Mathematics Zhejiang 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,.

loader
I am the owner, or an agent authorized to act on behalf of the owner, of the copyrighted work described.
capcha
Download Presentation

PowerPoint Slideshow about ' Chromatic Ramsey Number and Circular Chromatic Ramsey Number' - delila


An Image/Link below is provided (as is) to download presentation

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.


- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -
Presentation Transcript

Chromatic Ramsey Number and

Circular Chromatic Ramsey Number

Xuding Zhu

Department of Mathematics

Zhejiang 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.


Colour the edges of by red or blue,

there is either a red or a blue

Among 6 people,

There are 3 know each other,

or 3 do not know each other.


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.

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

Sufficiently large or complicated

General Ramsey Type Theorem:


A sufficiently largescale (or complicated) systemmust contains

an interesting sub-system.

“Complete disorder is impossible”


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.


Theorem [Ramsey, 1927]

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

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

whenever

the

elements

of

some

(sufficiently

large)

object

are

partitioned

into

a

finite

number

of

classes

(i.e.,

colored

with

a

finite

number

of

colors),

there

is

always

at

least

one

(color)

class

which

contains

all

the

elements

of

some

regular

structure.

When

this

is

the

case,

one

additionally

would

like

to

have

quantitative

estimates

of

what

“sufficiently

large”

means.

In

this

sense,

the

guiding

philosophy

of

Ramsey

theory

can

be

described

by

the

phrase:

“Complete

disorder

is

impossible”

.


Van der Waeden Theorem

For any partition of integers into finitely many parts,

one part contains arithematical progression of arbitrary

large length.

Regularity lemma

Erdos and Turan conjecture (1936)

Szemerédi's theorem (1975)

Every set of integers A with positive density contains

arithematical progressionof arbitrary length.

Harmonic analysis

Timonthy Gowers[2001] gave a proof

using both Fourier analysis and combinatorics.


Ramsey number R(3,k)

Furstenberg

[124]

gave

ergodic

theoretical

and

topological

dynamics

reformulations.


means.

For any 2-colouring of the edges of F with coloursredandblue,

there is a red copy ofGor a blue copy ofH.



1933, George Szekeres, Esther Klein, Paul Erdos

starting with a geometric problem, Szekeres re-discovered

Ramsey theorem, and proved


Szekere [1933]

Erdos [1946]

Erdos [1961]

Graver-Yackel [1968]

Ajtai-Komlos-Szemeredi

[1980]

Kim [1995]

Many sophisticated probabilistic tools are developed


George Szekere and Esther Klein married

lived together for 70 year,

died on the same day 2005.8.28, within one hour.





A sufficiently largescale (or complicated) systemmust contains

an interesting sub-system.

How to measure a system?

What is large scale?

What is complicated?

How to measure a graph?


Chromatic number

Circular chromatic number


G=(V,E): a graph

0

an integer

1

1

An k-colouring of G is

2

0

such that

A 3-colouring of


The chromatic number of G is


G=(V,E): a graph

0

a real number

1

an integer

1.5

A (circular)

k-colouring of G is

r-colouring of G is

An

2

0.5

A 2.5-coloring

such that


The circular chromatic number of G is

{ r: G has a circular r-colouring }

min

inf


f is k-colouring of G

f is a circular k-colouring of G

Therefore for any graph G,


0=r

0

r

1

4

2

3

|f(x)-f(y)|_r ≥ 1

x~y

p

p’

The distance between p, p’ in the circle is

f is a circular r-colouring if


and

Basic relation between

Circular chromatic number of a graph is a refinement

of its chromatic number.


Graph coloring is a model for resource distribution

Circular graph coloring is a model for resource distribution

of periodic nature.



If F has chromatic number , then there is a

2 edge colouring of F in which each monochromatic

subgraph has chromatic number n-1.

for any n-chromatic G.


If F has chromatic number , then there is a

2 edge colouring of F in which each monochromatic

subgraph has chromatic number n-1.

for any n-chromatic G.

Could be much larger


There are some upper bounds on

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

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

Attempts by Tardif, West, etc. on non-diagonal cases

of chromatic Ramsey numbers of graphs.

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

The conjecture is true (Zhu, 2011)


For any 2 edge-colouring of Kn, there is a monochromatic

graph which is a homomorphic image of G.



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.

The construction of G is easy:

Take all 2 edge colourings of

For each 2 edge colouring ci of , one of the monochromatic

subgraph, say Gi, , has chromatic number at least n.


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.

The construction of G is easy:

Take all 2 edge colourings of

For each 2 edge colouring of , one of the monochromatic

subgraph, say Gi, , has chromatic number at least n.


H

G

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.

?

The construction of G is easy:

Take all 2 edge colourings of

For each 2 edge colouring ci of , one of the monochromatic

subgraph, say Gi, , has chromatic number at least n.


H

G


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.

?

If Hedetniemi’s conjecture is true, then

Burr-Erdos-Lovasz conjecture is true.


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

integer linear programming


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

linear programming



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.

If Hedetniemi’s conjecture is true, then

Burr-Erdos-Lovasz conjecture is true.


To prove this conjecture for true,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.

The construction of G is easy:

Take all 2 edge colourings of

For each 2 edge colouring ci of , one of the monochromatic

subgraph, say Gi, , has chromatic number at least n.

fractional chromatic number > n-1


Theorem true,[Huajun Zhang, 2011]

If both G and H are vertex transitive, then

Fractional Hedetniemi’s conjecture

Theorem [Z, 2011]


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

dual problem

linear programming


The fractional chromatic number of G is obtained true,

by solving a linear programming problem

The fractional clique number of G is obtained

by solving its dual problem


Fractional Hedetniemi’s conjecture is true true,

Theorem [Z, 2010]

Easy!

Difficult!


Easy true,

Easy

Difficult


Difficult! true,

Easy!



is a refinement of ?

and

Basic relation between

is an approximation of


The reciprocal of is studied by computer ?

scientists as efficiency of a certain

scheduling method, in 1986.

Circular colouring is a good model for periodical

scheduling problems

There are many periodical scheduling problems in

computer sciences.


Theorem [Zhu, 2011] ?

No conjecture yet!


Using fractional version of Hedetniemi’s conjecture, ?

Jao-Tardif-West-Zhu proved in 2014


min ?

?

min

No !

[ Jao-Tardif-West-Zhu, 2014]




ad