Xuding Zhu

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Circular colouring of graphs

• Xuding Zhu
• Zhejiang Normal University

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A distributed computation problem
V a set of computers
D a set of data files
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a
b
e
c
d
If x y, then x and y cannot operate at the
same time.
If x y, then x and y must alternate their turns
in operation.
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Schedule the operating time of the
computers efficiently.
Efficiency proportion of computers operating
on the average.
The computer time is discrete time 0, 1, 2,
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1 colouring solution
Colour the vertices of G with k
colours.
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a
The efficiency is 1/3
0
2
e
b
1
1
0
d
c
Colour the graph with 3 colours
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4
At time
0
1
2
3
e
e
a c
a c
b d
b d
Operate machines
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In general, at time t, those vertices x
with f(x)t mod (k) operate.
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Computer scientists solution
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Better than the colouring solution
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If initially the keys are assigned as above,
then no computer can operate.
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Once an orientation is given, then the
scheduling is determined.
The problem of calculating the efficiency (for a
given orientation) is equivalent to a well
studied problem in computer science the
minimum cycle mean problem
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Theorem Barbosa et al. 1989
There is an initial assignment of keys such that
the scheduling derived from such an assignment is
optimal.
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Circular colouring of graphs
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G(V,E) a graph
0
an integer
1
1
An k-colouring of G is
2
0
such that
A 3-colouring of
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The chromatic number of G is
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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
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The circular chromatic number of G is
r G has a circular r-colouring
inf
min
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f is k-colouring of G
f is a circular k-colouring of G
Therefore for any graph G,
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p
p
The distance between p, p in the circle is
f is a circular r-colouring if
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Circular coloring method
Let r
Let f be a circular r-coloring of G
x operates at time k iff for some integer m
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4
0
5
3
1
2
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is an approximation of
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1. Vince, 1988.
2. star chromatic number

More than 300 papers published.
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Interesting questions for
are usually also interesting for
There are also questions that are not
interesting for , but interesting for
For the study of , one may need to
sharpen the tools used in the study of
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Questions interesting for both and
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there is a graph G with
Answer Vince 1988
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there is a graph G with
Answer (Erdos classical result) all positive
integers.
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there is a graph G with
Answer Zhu, 1996
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there is a graph G with
Four Colour Theorem
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there is a graph G with
Four Colour Theorem implies
Answer Moser, Zhu, 1997
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there is a graph G with
Hadwiger Conjecture
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Hadwiger Conjecture implies
Answer Liaw-Pan-Zhu, 2003
Answer Hell- Zhu, 2000
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Hadwiger Conjecture implies
Answer Liaw-Pan-Zhu, 2003
Answer Hell- Zhu, 2000
Pan- Zhu, 2004
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there is a graph G with
Trivially all positive integers
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there is a graph G with
We know very little
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What happens in the interval 3,4?
Theorem Afshani-Ghandehari-Ghandehari-Hatami-Tuss
erkani-Zhu,2005
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What happens in the interval 3,4?
Maybe it will look like the interval 2,3
Gaps everywhere ?
NO!
Theorem Afshani-Ghandehari-Ghandehari-Hatami-Tuss
erkani-Zhu,2005
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What happens in the interval 3,4?
Theorem Lukotka-Mazak,2010
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Theorem Lukotka-Mazak,2010
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Theorem Lin-Wong-Zhu,2013
Theorem Lukotka-Mazak,2010
Theorem Lin-Wong-Zhu,2013
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For the study of , one may need to
sharpen the tools used in the study of
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A powerful tool in the study of list colouring
graphs is Combinatorial
Nullstellensatz
Give G an arbitrary orientation.
Find a proper colouring find a nonzero
assignment to a polynomial
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What is the polynomial for circular colouring?
Give G an arbitrary orientation.
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Colors assigned to adjacent vertices have
circular distance at least q
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Colors assigned to adjacent vertices have
circular distance at least q
1
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What is the polynomial for circular colouring?
Give G an arbitrary orientation.
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Theorem Alon-Tarsi
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Theorem Norine-Wong-Zhu, 2008
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Theorem Norine-Wong-Z, (JGT 2008)
q1 case was proved by Alon-Tarsi in 1992.
CorollaryNorine Even cycle are circular
2-choosable.
The only known proof uses combinatorial
nullstellensatz
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Circular perfect graphs
edge-preserving
Such a mapping is a homomorphism from G to H
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clique number
For a G, the chromatic number of G is
max
G
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A graph G is perfect if for every induced
subgraph H of G,
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be the graph with vertex set
For
let
V0, 1, , p-1
ij
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The chain
is extended to a dense chain.
G
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The circular chromatic number
min
inf
G
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clique
The circular chromatic number
min
max
G
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A graph G is perfect if for every induced
subgraph H of G,
A graph G is circular perfect if for every
induced subgraph H of G,
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A graph G is perfect if for every induced
subgraph H of G,
A graph G is circular perfect if for every
induced subgraph H of G,
Theorem Grotschel-Lovasz-Schrijver, 1981
For perfect graphs, the chromatic number is
computable in polynomial time.
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A graph G is perfect if for every induced
subgraph H of G,
A graph G is circular perfect if for every
induced subgraph H of G,
Theorem Bachoc Pecher Thiery, 2011
Theorem Grotschel-Lovasz-Schrijver, 1981
circular
circular
For perfect graphs, the chromatic number is
computable in polynomial time.
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A key step in the proof is calculating the Lovasz
theta number of circular cliques and their
complements.
Theorem Bachoc Pecher Thiery, 2011
Theorem Grotschel-Lovasz-Schrijver, 1981
circular
circular
For perfect graphs, the chromatic number is
computable in polynomial time.
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A key step in the proof is calculating the Lovasz
theta number of circular cliques and their
complements.
Theorem Bachoc Pecher Thiery, 2011
Theorem Grotschel-Lovasz-Schrijver, 1981
circular
circular
For perfect graphs, the chromatic number is
computable in polynomial time.
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A key step in the proof is calculating the Lovasz
theta number of circular cliques and their
complements.
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There are very few families of graphs for
which the theta number is known.
Theorem Bachoc Pecher Thiery, 2011
Theorem Grotschel-Lovasz-Schrijver, 1981
circular
circular
For perfect graphs, the chromatic number is
computable in polynomial time.
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Kneser graph KG(n,k)
Vertex set all k-subsets of 1,2,,n
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45
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Petersen graph
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25
13
KG(5,2)
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14
23
15
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There is an easy (n-2k2)-colouring of KG(n,k)
For i1,2,, n-2k1,
k-subsets with minimum element i is coloured
by colour i.
Other k-subsets are contained in n-2k2,,n and
are coloured by colour n-2k2.
Kneser Conjecture 1955
Lovasz Theorem 1978
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There is an easy (n-2k2)-colouring of KG(n,k)
For i1,2,, n-2k1,
k-subsets with minimum element i is coloured
by colour i.
Other k-subsets are contained in n-2k2,,n and
are coloured by colour n-2k2.
Johnson-Holroyd-Stahl Conjecture 1997
Kneser Conjecture 1955
Lovasz Theorem 1978
Chen Theorem 2011
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Lovasz Theorem
For any (n-2k2)-colouring c of KG(n,k), each
colour class is non-empty
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Alternative Kneser Colouring Theorem Chen, 2011
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Alternative Kneser Colouring Theorem Chen, 2011
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Alternative Kneser Colouring Theorem Chen, 2011
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Alternative Kneser Colouring Theorem Chen, 2011
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Alternative Kneser Colouring Theorem Chen, 2011
1
1
2
2
3
3
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4
5
5
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Alternative Kneser Colouring Theorem Chen, 2011
Chang-Liu-Zhu, A simple proof (2012)

For any (n-2k2)-colouring c of KG(n,k), there
exists a colourful copy of ,
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Thank you!
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Thank you!
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Proof (Chang-Liu-Z, 2012)
The proof uses Ky Fans Lemma from algebraic
topology.
Consider the boundary of the
n-dimensional cube
which is an (n-1)-dimensional sphere.
We construct a triangulation of
as follows
The vertices of the triangulation are
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The vertices of the triangulation are
A set of distinct vertices form a simplex if the
vertices can be ordered
so that
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Examples
n2
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Examples
The vertices are
n2
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Examples
The vertices are
n2
Each vertex is a 0-dimensional simplex (0-simplex)
There are 8 1-simplices
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Examples
n3
A 2-simplex
There are 48 2-simplices
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Ky Fan Lemma (special case of the lemma needed)
Let
be such that
Then there is an odd number of (n-1)-simplices
whose vertices are labeled with
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Assume c is a (n-2k2)-colouring of KG(n,k).
Associate to c a labeling of
as follows
For
let