Painting game on graphs

Xuding Zhu

Zhejiang Normal University

2014.05.28

8th Shanghai Conference on Combinatorics

There are six teams, each needs to competewith all the others.

Each team can play one game per day

How many days are needed to scheduleall the games?

Answer: 5 days

A scheduling problem:

1st day

2nd day

3rd day

4th day

5th day

5)(' 6 K

12)(' 2 nK n

This is an edge colouring problem.

Each day is a colour.

There are six teams, each needs to competewith all the others.

Each team can play one game per day

How many days are needed to scheduleall the games?

Answer: 5 days

Each team can choose one day off

7 days are enough

A scheduling problem:

7 days are needed

Each edge misses at most 2 colours

There are 7 colours

Each edge has 5 permissible colours

5)('ch 6 K

I do not know any easy proof

List colouring conjecture:

)(')('ch GG For any graph G,

However, the conjecture remains open for nK2

Haggkvist-Janssen (1997) nKn )('ch

n)ch'(K)'(Kn- nn 212 22

There are six teams, each needs to competewith all the others.

Each team can play one game per day

How many days are needed to scheduleall the games?

Answer: 5 days

Each team can choose one day off

7 days are enough

The choices are madebefore the scheduling

A scheduling problem

There are six teams, each needs to competewith all the others.

Each team can play one game per day

How many days are needed to scheduleall the games?

7 days are enough

Each team can choose one day off

A scheduling problem

is allowed not to show up for one day

On each day, we know which teams haven’t shown up today

but we do not know which teams will not show up tomorrow

We need to schedule the games for today

On-line list colouring of graphs

We start colouring the graph

before having the full information of the list

f-painting game (on-line list colouring game) on G

,2,1,0)( : GVf

is the number of permissible colours for x)(xf

Two Players:

Lister Painter

Each vertex v is given f(v) tokens.

Each token represents a permissible colour.

But we do not know yet what is the colour.

Reveal the list Colour vertices

At round i

is the set of vertices which has colour i asa permissible colour.iV

Painter chooses an independent subset of iI iV

vertices in are coloured by colour i.iI

Lister choose a set of uncoloured vertices, removesone token from each vertex of

iV

iV

If at the end of some round, there is an uncoloredvertex with no tokens left, then Lister wins.

If all vertices are coloured thenPainter wins the game.

G is f-paintable if Painter has a winning strategy for the f-painting game.

G is k-paintable if G is f-paintable for f(x)=k for every x.

The paint number of G is the minimum k for which G is k-paintable.

)(chp G

choice number choosable- is :min)(ch kGkG

)(ch)(chp GG

Painter start colouring the graph

after having the full information of the list

List colouring:On-line list colouring:

before

4,2,2

Theorem [Erdos-Rubin-Taylor (1979)]

n2,2,2 is 2-choosable.

4,2,2 is not 2-paintable

4,2,2

1

12

2 3

334

4 5

5Lister wins the game

4,2,2 is not 2-paintable

Theorem [Erdos-Rubin-Taylor,1979]

A connected graph G is 2-choosable if and only if its core is

1K ornC2 p2,2,2or

p2,2,2However, if p>1, then is not 2-paintable.

Theorem [Zhu,2009]

A connected graph G is 2-paintable if and only if its core is

1K ornC2 2,2,2or

A recursive definition of f-paintable

Assume . Then G is f-paintable, if ,2,1,0)( : GVf

(1) )(GV or

(2).paintable is

, set tindependen ),(

)(f-δG-I

XI GVX

X

XX of function sticcharacteri :

For any question about list colouring,we can ask the same question for on-line list colouring

Planar graphs and locally planar graphs

Chromatic-paintable graphs

Partial painting game

Complete bipartite graphs

b-tuple painting game and fractional paint number

For any result about list colouring,we can ask whether it holds for on-line list colouring

Some upper bounds for are automatically upper bounds for

)(chG)(chp G

Theorem [Galvin,1995] If G is bipartite, then )()(' GGp

Upper bounds for ch(G) proved by kernel method are also upperbounds for )(chp G

Upper bounds for ch(G) proved by Combinatorial Nullstellensatz are also upper bounds for

paintable1 is then

,

whichfor norientatio an has If

2009] [Schauz, Theorem

)(dG

|OE(D)| |EE(D)|

DG

D

)(chp G

Brooks’ Theorem

choosable is then, , If 12 G CKG knpaintable

[Hladky-Kral-Schauz,2010]

Upper bounds for ch(G) proved by induction

Theorem [Thomassen, 1995] Every planar graph is 5-choosable

Planar graphs

[ Schauz,2009 ] paintable

Thomassen proves a stronger result:

),,,(C

cycle boundary withgraph planar a is Assume

21 kvvv

G

,)( for 5)( and

,3 for 3)( ,1)()(v If 21

CGVvvf

kivfvff i

choosable.- is then fG paintable

Basically, Thomassen’s proof works for f-paintable.

Locally planar

embedded in a surface G

contractible

non-contractible

edge-width of G

length of shortest non-contractible cycle

edge-width is large torus ：

Theorem [Thomassen, 1993] For any surface , there is a constant , any G embedded in with edge-width > is 5-colourable.

w w

Locally planar

embedded in a surface G

contractible

non-contractible

edge-width of G

length of shortest non-contractible cycle

edge-width is large torus ：

Theorem [Thomassen, 1993] For any surface , there is a constant , any G embedded in with edge-width > is 5-colourable.

w w

DeVos-Kawarabayashi-Mohor 2008

choosable

Han-Zhu 2014+

paintable

Find a subgraph H

G-H is planar Each piece in H is planar

Apply strategy for planar graphs on the pieces of H and on G-H, one by one

+ some other nice properties

Chromatic-paintable graphs

A graph G is chromatic choosable if )()(ch GG

Conjecture: Line graphs are chromatic choosable.

Conjecture: Claw-free graphs are chromatic choosable.

Conjecture: Total graphs are chromatic choosable.

Ohba Conjecture: Graphs G with are chromatic choosable.

1)(2|)(| GGV Theorem [Noel-Reed-Wu,2013]

paintable )()( GGchp

paintable

paintable

paintable

Conjecture: Graph squares are chromatic choosable. [Kim-Park,2013]

paintable

A graph G is chromatic choosable if )()(ch GG

Conjecture: Line graphs are chromatic choosable.

Conjecture: Claw-free graphs are chromatic choosable.

Conjecture: Total graphs are chromatic choosable.

Ohba Conjecture: Graphs G with are chromatic choosable.

1)(2|)(| GGV

paintable )()( GGchp

paintable

paintable

paintable

Conjecture: Graph squares are chromatic choosable. [Kim-Park,2013]

paintableNO! Question

3,2,2K is not 3-paintable.

33

33

333

Lister

33

33

333

Lister

323233

Painter

232333

33

33

333

Lister

323233

Painter

232333

Lister

33

33

333

Lister

323233

Painter

232333

Lister

33

33

333

Lister

323233

Painter

232333

Lister

Painter

13222

23222

31322

33

33

333

Lister

323233

Painter

232333

Lister

Painter

13222

23222

31322

Lister

33

33

333

Lister

323233

Painter

232333

Lister

Painter

13222

23222

31322

Lister

3111

Painter Lose

2112

Painter Lose

2311

Painter Lose

3,2,2K

is not 3-paintable

{123}

{1}{2}{3}

Theorem [Kim-Kwon-Liu-Zhu,2012]

For k>1, is not (k+1)-paintable.3,2 kK

Ohba Conjecture: Graphs G with are chromatic choosable.

1)(2|)(| GGV paintable

On-line version Huang-Wong-Zhu 2011

To prove this conjecture, we only need to considercomplete multipartite graphs.

1v 2v

3v 4v

12 nv nv2

Theorem [Huang-Wong-Zhu,2011]

nK 2 is n-paintable

The first proof is by usingCombinatorial Nullstellensatz

A second proof gives a simple winning strategy for Painter

The proof uses induction.

Theorem [Kozik-Micek-Zhu,2014]

On-line Ohba conjecture is true for graphswith independence number at most 3.

The key in proving this theorem is to find a “good” technical statement that can be proved by induction.

1 size of parts

2 size of parts

3 size of parts

Partition of the partsinto four classes

2or 1 size of parts

ordered

ordered

1 size of parts

2 size of parts

3 size of parts

2or 1 size of parts

ordered

ordered

1A

2A

1kA

1S

2S

sS

2k

3k

iB

iC

||)(1

ij

jS Siv

vAi For ikkvf 32)(

vuBi ,For 32)( kkvf

|)(|)()( GVufvf

iSvFor )(2)( 321 ivkkkvf S

wvuCi ,,For 32)( kkvf

1|)(|)()( GVufvf

3211|)(|)()()( kkkGVwfufvf

G is f-paintable

Theorem [Kozik-Micek-Zhu,2014]

On-line Ohba conjecture is true for graphswith independence number at most 3.

Theorem [Chang-Chen-Guo-Huang,2014+]

)(432

|| ,)( If2

2

Gmmmm

VmG

paintable- chromatic is thenG

paintable)(47

||,4)( GVg

Theorem [Erdos,1964] then,n vertices has and bipartite isG If

Theorem[Zhu,2009]

If G is bipartite and has n vertices, then

1log)( 2 nGchp

1log)(ch 2 nG

Complete bipartite graphs

probabilistic proof

Painter colours , double the weight of each vertex in

iVAiVB

A

B

Initially, each vertex x has weight w(x)=1

Assume Lister has given set iV

If )()( ii VBwVAw

If x has permissible colours, Painter will be able to colour it.

1log2 n

A

B

The total weight of uncoloured vertices is not increased.

If a vertex is given a permissible colour but is not coloured by that colour, then its weight doubles.

If x has been given k permissible colours, but remains uncoloured,then

kxw 2)( nxw k 2)( nk 2log

Theorem [Carraher-Loeb-Mahoney-Puleo-Tsai-West]

rkK ,

XY

rYXk ||||

k

,...,2,1: Vf

k

k

kt

2t1t

k

ii

rk

tr

K

1

,

iff

paintable-f is

3-choosable complete bipartite graphs

Theorem

iff q)(p choosable-3 is , qpK

26,3 qp

18,4 qp

8,5 qp

7,6 qp

Mahadev-Roberts-Santhanakrishnan, 1991

Furedi-Shende-Tesman, 1995

O-Donnel,1997

3-paintable complete bipartite graphs

Theorem [Chang-Zhu,2013]

iff q)(p paintable-3 is , qpK

26,3 qp

11,4 qp

8,5 qp

7,6 qp

Theorem [Erdos, 1964]

noK nn 2, log))1(1()(ch

noKch nn 2,p log))1(1()(

)( ,p nnKch)( ,nnKch and

Theorem [Zhu, 2010]

Erdos-Lovasz Conjecture

nonK nn 222, loglog))1(1(log)(ch

Maybe

?loglog)1(log)( 222,p nonKch nn

nonKnon nn 222,222 loglog))1(21

(log)(chloglog))1(2(log

2log)(chloglogloglogloglog 2,p222222 nKnnn nn

Theorem [RadhaKrishnan-Srinivasan,2000]

nonK nn 222, loglog)1(21

log)(ch

Probabilistic mothed.

)( ,p nnKch)( ,nnKch and

Theorem [Duray-Gutowksi-Kozik,2014+]

nnKch nn 222,p logloglog)(

Theorem [Gerbner-Vizer, 2014+]

nnnKch nn 222222,p loglogloglogloglog)(

1c constant some for logloglog)( 222,p ncnKch nn ?

b-tuple list colouring

b-tuple on-line list colouring

G is (a,b)-choosable

if |L(v)|=a for each vertex v, then there is a b-tuple L-colouring.

If each vertex has a tokens, then Painter has a strategy to colour each vertex a set of b colours.

(a,b)-choosable

Conjecture [Erdos-Rubin-Taylor]

(am,bm)-choosable

On-line version

(a,b)-paintable (am,bm)-paintable

Theorem [Tuza-Voigt, 1996]

2-choosable (2m,m)-choosable

Theorem [Mahoney-Meng-Zhu, 2014]

2-paintable (2m,m)-paintable

colourable-b)(a, is G :inf)(ba

Gf

choosable-b)(a, is G :inf)(ba

Gchf

paintable-b)(a, is G :inf)(ba

Gchfp

)()( GchG ff

Theorem

)()( GchG fPf

[Alon-Tuza-Voigt, 1997] [Gutowski, 2011]

Infimum attained Infimum not attained

)()( GchG ff

Theorem

)()( GchG fPf

[Alon-Tuza-Voigt, 1997] [Gutowski, 2011]

Infimum attained Infimum not attained

Every bipartite graph is (2m,m)-choosable for some m

Theorem [Mahoney-Meng-Zhu,2014]

For any m, a connected graph G is (2m,m)-paintable if and only if its core is

1K or nC2 2,2,2or

Partial painting game

Partial f-painting game on G

same as the f-painting game, except that Painter’s goal is not to colour all the vertices, but to colour as many vertices as possible.

Fact:

least at colour

can one colours, kk' using then ,)( If kG

V kk'

vertices.

Conjecture [Albertson]:

least at colour can one colours,

epermissibl kk' has vertex each ,)(h If kGc

V kk'

vertices.

Conjecture [Zhu, 2009]:

least at colour to strategy a has Painter then

tokens kk' has vertex each ,)(h If P kGc

V kk'

vertices.

Conjecture [Zhu, 2009]:

least at colour to strategy a has Painter then

tokens kk' has vertex each ,)(h If P kGc

V kk'

vertices.

Theorem [Wong-Zhu,2013]

76

Question: Can the difference be arbitrarily large ?

)(ch)(p GGch

Nine Dragon TreeNine Dragon Tree

Thank you

An easy but useful lemma:

Corollary: If G is k-degenerate, then G is (k+1)-paintable

If f(v) >d(v), then G is f-paintable iff G-v is f-paintable.

Painter uses his winning strategy on G-v.

Colour v only if v is marked (i.e., the current colour is permissible for v), and none of its neighbours used the current colour.

v will be coloured when all its tokens are used gone, if not earlier.

2v

kv1v

Case 1

u

w

2v

kv1v

1G2G

iV

)( 11 GVVV ii

Painter apply his winning strategy on with 1G

, obtain independent set 11ii VI

Case 1

wuGVVwuIV iii ,)(, 212

Then Painter apply his winning strategy on with 2G

obtain independent set 22ii VI

u

w

221 in , of

role the play ,

Gvv

wu

21iii III

2v

kv1v

Case 2

iV

'iV defined as follows:

kiii vVVVv ' then , If 1

][' then , and If 1 kGiiiki vNVVVvVv

ii VV ' Otherwise

Painter apply his winning strategy on with input kvG 'iV

obtains 'iI add if possible. kv