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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,. - PowerPoint PPT Presentation
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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
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