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.
Among 6 people,
There are 3 know each other,
or 3 do not know each other.
Colour the edges of K 6 by red or blue,
there is either a red K 3 or a blue K 3
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:
For `any’ systems Gi (1 i  k) , there exists
a system F such that if `elements’ of F are partitioned
into k parts, then for some i, the ith part contains
Gi as a subsystem.
A sufficiently large scale (or complicated) system must 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]
t ,n ,k ,
m
For a set M with | M | m,
If the k-tuples M are t-colored, then
M '  M , | M ' | n
all the k-tuples of M’ having the same color.
k  1 : pigeonholeprinciple
k  2 : colouringtheedges of a completegraph
Van der Waeden Theorem
For any partition of integers into finitely many parts,
one part contains arithematical progression of arbitrary
large length.
Szemerédi's
theorem
(1975)
Erdos
and Turan
conjecture
(1936)
Regularity lemma
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.
Harmonic analysis
Ramsey number R(3,k)
F  (G, H )
means.
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 6  ( K3 , K3 )
K 5   ( K3 , K3 )
The Ramsey number of (G,H) is
R(G,H) minn : Kn  (G, H )
R( Kk , Kl )  R(k , l )
R( 3, k)  minn : Kn  ( K3 , Kk )
1933, George Szekeres, Esther Klein, Paul Erdos
starting with a geometric problem, Szekeres re-discovered
Ramsey theorem, and proved
2
R( 3, k)  k
Szekere [1933]
Erdos [1946]
Erdos [1961]
R( 3, k)  k 2
  k 3/2 

R( 3, k)   

  lnk  


  k 2 

R( 3, k)   

  lnk  


Graver-Yackel [1968]
 ln ln k 
R( 3, k)  O k 2

ln k 

Ajtai-Komlos-Szemeredi
[1980]
 k2 

R( 3, k)  O
 ln k 
Kim [1995]
 k2 

R( 3, k)  
 ln k 
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.
R(k,l) minn : Kn  ( Kk , Kl )
Bounds for R(k,l)
k
l 3
4
5
6
7
8
3
6
9
14
18
23
28
18
25
36
41
49
61
58
84
43
49
58
87
80
143
101
216
102
165
113
298
132
495
205
540
217
1031
4
5
6
7
8
282
1870
Bounds for R(k,l)
k
l 3
4
5
6
7
8
3
6
9
14
18
23
28
18
25
36
41
49
61
58
84
43
49
58
87
80
143
101
216
102
165
113
298
132
495
205
540
217
1031
4
5
6
7
8
282
1870
Bounds for R(k,l)
k
l 3
4
5
6
7
8
3
6
9
14
18
23
28
18
25
36
41
49
61
58
84
43
49
58
87
80
143
101
216
102
165
113
298
132
495
205
540
217
1031
4
5
6
7
8
282
1870
A sufficiently large scale (or complicated) system must contains
an interesting sub-system.
How to measure a system?
What is large scale?
What is complicated?
How to measure a graph?
T heRamsey number of (G, H) is
R(G,H) min n : K n  (G, H )
T heRamsey number of (G, H) is
R(G,H) min | V ( F ) |: F  (G, H )
T heSize Ramsey number of (G, H) is
RE(G,H) min | E ( F ) |: F  (G, H )
T hemax - degree - Ramsey number of (G, H) is
R(G,H) min | ( F ) |: F  (G, H )
T hechromaticRamsey number of (G, H) is
R (G,H) min  ( F ) : F  (G, H )
T hecircular chromaticRamsey number of (G, H) is
Rc (G,H) inf   c ( F ) : F  (G, H )
Chromatic number
Circular chromatic number
G=(V,E): a graph
k  1 : an integer
An k-colouring of G is
f : V  0,1,..., k  1
such that
x ~ y  f ( x)  f ( y )
0
1
1
2
0
k 3
A 3-colouring of C5
The chromatic number of G is
 (G)  mink : G has a k - colouring
G=(V,E): a graph
kr  1 :
a real
number
an
integer
A (circular)
k-colouring
r-colouring
of G is of G is
An
r ) k  1
f : V  [[000,,1,k,...,
0
1
1.5
0.5
2
A 2.5-coloring
such that x ~ y 
f ( x) 1f(|yf) ( x) 
 f ( y1) || rf (x1)  f ( y ) | k  1
The circular chromatic number of G is
 c (G)  min
inf { r: G has a circular r-colouring }
f is k-colouring of G
f is a circular k-colouring of G
Therefore for any graph G,
c (G)   (G)
0=r
p
p’
0
1
r
4
2
3
The distance between p, p’ in the circle is
| p - p'|r  min | p  p'|, r  | p  p'| 
f is a circular r-colouring if
x~y
|f(x)-f(y)|_r ≥ 1
Basic relation between
 (G ) and  c (G )
 (G)  1  c (G)   (G).
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.
T hechromaticRamsey number of (G, H) is
R (G,H) min  ( F ) : F  (G, H )
R(G)  R (G, G)
Introduced by Burr-Erdos-Lovasz in 1976
R(G, H)  R(G, H )
R(Kk ,Kl )  R(k , l )
2
(
n

1
)
If F has chromatic number
, then there is a
2 edge colouring of F in which each monochromatic
subgraph has chromatic number n-1.
F  (G, G)
n4
for any n-chromatic G.
2
(
n

1
)
If F has chromatic number
, then there is a
2 edge colouring of F in which each monochromatic
subgraph has chromatic number n-1.
F  (G, G)
for any n-chromatic G.
Could be much larger
Observation : If  (G)  n, then R (G)  (n 1)2 1
Conjecture[Burr - Erdos- Lovasz,1976]: For each n,
thereis a graph G with  (G)  n and R (G)  (n  1) 2  1
The conjecture is true for n=3,4 (Burr-Erdos-Lovasz, 1976)
The conjecture is true for n=5 (Zhu, 1992)
There are some upper bounds on minR  (G) :  (G)  n
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)
Conjecture[Burr - Erdos- Lovasz,1976]: For each n,
2
2
thereis amin
graph
G
with

(
G
)

n
and
R
(
G
)

(
n

1
)
1
R  (G) :  (G)  n (n 1) 1 
Lemma[Burr - Erdos- Lovasz]
R (G)  n  K n  hom(G)
For any 2 edge-colouring of Kn, there is a monochromatic
graph which is a homomorphic image of G.
Graph homomorphism = edge preserving map
G
H
To prove Burr-Erdos-Lovasz conjecture for n, we need to construct
an n-chromatic graph G, so that any 2 edge colouring of K ( n1)2 1
has a monochromatic subgraph which is a homomorphic image of G.
The construction of G is easy:
Take all 2 edge colourings of
K( n1)2 1
c1 , c2 ,, cm
For each 2 edge colouring ci of K ( n1)2 1 , one of the monochromatic
Gi , has chromatic number at least n.
subgraph, say Gi,
Conjecture[Burr - Erdos- Lovasz,1976]: For each n,
thereis a graph G with  (G)  n and R (G)  (n  1) 2  1
To prove this conjecture for n, we need to construct an
n-chromatic graph G, so that any 2 edge colouring of K ( n1)2 1
has a monochromatic subgraph which is a homomorphic image of G.
The construction of G is easy:
Take all 2 edge colourings of
K( n1)2 1
c1 , c2 ,, cm
For each 2 edge colouring ci of K ( n1)2 1 , one of the monochromatic
Gi , has chromatic number at least n.
subgraph, say Gi,
G  G1  G2  Gm
GH
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 K ( n1)2 1
has a monochromatic subgraph which is a homomorphic image of G.
?
The construction of G is easy:
Take all 2 edge colourings of
K( n1)2 1
c1 , c2 ,, cm
For each 2 edge colouring ci of K ( n1)2 1 , one of the monochromatic
Gi , has chromatic number at least n.
subgraph, say Gi,
G  G1  G2  Gm
Each Gi is a homomorphi
c image of G
H
G
Conjecture[Hedetniemi, 1966]:
 (G  H )  min (G),  ( H )
To prove this conjecture for n, we need to construct an
n-chromatic graph G, so that any 2 edge colouring of K ( n1)2 1
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.
(G) : familyof independent sets of G
 (G)  min
 ( X )
X  ( G )
 : (G)  0,1
 ( X )  1
vX , X  (G )
integer linear programming
Af k-colouring
(G) : fractional
chromatic
of G sets.
of G partition
V(G) number
into k independent
(G) : familyof independent sets of G
f (G
(G))  min
 ( X )
X  ( G )
 : (G)  [0,1]
 ( X )  1
vX , X  (G )
linear programming
 f (G)   (G)
Conjecture[Hedetniemi, 1966]:
 (G  H )  min (G),  ( H )
Fractional Hedetniemi’s conjecture
Conjecture[Z, 2002]:
 f (G  H )  min f (G),  f ( H )
To prove this conjecture for n, we need to construct an
n-chromatic graph G, so that any 2 edge colouring of K ( n1)2 1
has a monochromatic subgraph which is a homomorphic image of G.
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 K ( n1)2 1
has a monochromatic subgraph which is a homomorphic image of G.
The construction of G is easy:
Take all 2 edge colourings of
K( n1)2 1
c1 , c2 ,, cm
For each 2 edge colouring ci of K ( n1)2 1 , one of the monochromatic
Gi , has chromatic
subgraph, say Gi,
at least
n. > n-1
fractionalnumber
chromatic
number
G  G1  G2  Gm  f (G)  n 1
 (G)   f (G)  n 1
Each Gi is a homomorphi
c image of G
Conjecture[Hedetniemi, 1966]:
 (G  H )  min (G),  ( H )
Fractional Hedetniemi’s conjecture
Conjecture
[Z,
2002]:
Theorem [Z,
2011]
 f (G  H )  min f (G),  f ( H )
Theorem [Huajun Zhang, 2011]
Conjecture
: transitive, then
If both G and[Z,
H 2002]
are vertex
 f (G  H )  min f (G),  f ( H )

Af k-colouring
(G) : fractional
clique
chromatic
number
number
Gof G sets.
of G partition
V(G)
into kofindependent
(G) : familyof independent sets of G
f (G
(G))  min
 ( X )
X  ( G )
 : (G)  [0,1]
 ( X )  1
vX , X  (G )
linear programming
 f (G)  max
 (v)
vV ( G )
 : V (G)  [0,1]
 (v)  1
vX , X  ( G )
dual problem
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
 f (G)   f (G)
Fractional Hedetniemi’s conjecture is true
Conjecture
[Z,
2002]:
Theorem [Z,
2010]

 f (G  H )  min
 f (G),
 f (H )
Pr oof sketch :
 f (G  H )  min f (G),  f (H )
 f (G  H )  min f (G),  f (H )
Easy!
Difficult!
 f (G  H )  min f (G), f (H )
suffices to constructa fractionalclique of G  H
with totalweight min f (G), f ( H )
Easy
Easy
Difficult
suffices to constructa fractionalclique of G  H
with totalweight min f (G), f ( H )
g : V(G)  [ 0,1], a maximumfractionalclique of G
h : V ( H )  [0,1], a maximumfractionalclique of H
 : V (G  H )  [0,1], defined as
g(x)h(y)
(x,y) 
maxω f (G),ω f (H)
is a fractionalclique of G  H
Difficult!
with totalweight min f (G), f ( H )
 independent set U of G  H ,
 g ( x)h( y)  max
( x , y )U
f
(G), f ( H )
Easy!
T hecircular chromaticRamsey number of (G, H) is
Rc (G,H) inf   c ( F ) : F  (G, H )
Rc (G)  Rc (G, G)
What is the relation between
Rc (G) and
 c(G) ?
Basic relation between
 (G ) and  c (G )
 (G)  1  c (G)   (G).
c (G)   (G).
 c (G ) is a refinement of  (G )
 (G ) is an approximation of  c (G )
Circular colouring is a good model for periodical
scheduling problems
There are many periodical scheduling problems in
computer sciences.
The reciprocal of  c (G ). is studied by computer
scientists as efficiency of a certain
scheduling method, in 1986.
R (n)  minR (G) :  (G)  n
2
Conjecture
[Burr
Erdos
Lovasz,
1976]
:
R
(
n
)

(
n

1
)
1

Theorem [Zhu, 2011]


Rc ( z)  inf Rc (G) : c (G)  z
Rc ( z)  ?
No conjecture yet!
 f (G)  c (G)   (G).
 (G)  c (G)  1.
 (G)   f (G) can be arbitrarily large.
Using fractional version of Hedetniemi’s conjecture,
Jao-Tardif-West-Zhu proved in 2014
If k  2 is an integer, thenRc (k )  k (k 1)
Rc (2)  2(2 1)
Rc (K3 )  3(3 1)
Rc (3)  3(3 1) ?
c (G) can be real rational 2
Whatare thepossible valuesof Rx (G) ?
c(G)  min
inf  r : G has a circular r - colouring
Rc (G)  min
inf
 c ( F ) :
F  (G, G) ?
No !
[ Jao-Tardif-West-Zhu, 2014]
Some other results by Jao-Tardif-West-Zhu, 2014
5
If 2  z  , then R c ( z )  4
2
Rc ( K3 )  6
Rc ( K3 , K4 )  9
14
 R c (C3 , C5 )  5
3
9
4  R c (C3 , C7 ) 
2
謝謝