on-line list colouring of graphs

On-line list colouring of graphs

Xuding Zhu

Zhejiang Normal University

Suppose G is a graph

A list assignment L assigns to each vertex x a set L(x) of permissible colours.

An L-colouring h of G assigns to each vertex x a colour

)()( xLxh such that for every edge xy )()( yhxh

choice number

colouring-an has then

,every for if choosable- is

LG

xk|L(x)|kG

choosable- is :min)(ch kGkG

Given a vertex x, L(x) tells us which colours are permissible.

Alternately, given a colour i, one can ask which vertices have i as a permissible colour.

A list assignment L can be given as

Given a colour i, is the set of vertices having i as a permissiblecolour.

iL

mLLL ,,, 21

An L-colouring is a family of independent subsets mAAA ,,, 21

ii LA and mAAAV 21

on-line f-list colouring game on G

played by Alice and Bob

At round i, Alice choose a set of uncoloured vertices. iV

,2,1,0)( : GVf

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

is the number of permissible colours for x)(xf

Bob chooses an independent subset of and colour vertices in by colour i.

iI iV

iI

Alice wins the game if there is a vertex x, which has been given f(x) permissible colours and remains uncoloured.

Otherwise, eventually all vertices are coloured and Bob wins the game.

G is on-line f-choosable if Bob has a winning strategy for the on-line f-list colouring game.

G is on-line k-choosable if G is on-line f-choosable for f(x)=k for every x.

The on-line choice number of G is the minimum k for which G is on-line k-choosable.

)(chOL G

)(ch)(chOL GG

4,2,2

Theorem [Erdos-Rubin-Taylor (1979)]

n2,2,2 is 2-choosable.

4,2,2 is not on-line 2-choosable

4,2,2

1

1

4,2,2 is not on-line 2-choosable

22 3

334

4 5

5Alice wins the game

Question: Can the difference be arbitrarily large ?

)(ch)(chOL GG

Question: Can the ratio be arbitrarily large ?

)(ch/)(chOL GG

Most upper bounds for choice number are also upper bounds for on-line choice number.

Currently used method in proving upper bounds for choice number

Kernel method

Induction Some works for on-line choice number,

Combinatorial Nullstellensatz

Theorem [Schauz,2009] For planar G, .5)(chOL G

Theorem [Chung-Z,2011] For planar G, triangle free + no 4-cycleadjacent to a 4-cycle or a 5-cycle, .3)(chOL G

Theorem [Schauz,2009] Upper bounds for ch(G) proved byCombinatorial nullstellensatz works for on-line choice number

Theorem [Schauz,2009] Upper bounds for ch(G) proved byCombinatorial nullstellensatz works for on-line choice number

Theorem [Schauz,2009]

If G has an orientation D with

|)(||)(| DEODEE

then G is on-line choosable)1( Dd

The proof is by induction (no polynomial is involved).

Probabilistic method Does not work for on-line choice number

Theorem [Alon, 1992] |)(|ln)()(ch GVGcG

The proof is by probabilistic method

1|)(|ln)()(chOL GVGG Theorem[Z,2009]

Proof: If G is bipartite and has n vertices, then

1log)(ch 2OL nG

948c

If a vertex x has permissible colours, Bob will be able to colour it.

1log2 n

Bob colours , double the weight of each vertex in

iVAiVB

A

B

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

Assume Alice has given set iV

If )()( ii VBwVAw

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 it weight doubles.

If a vertex x has given k permissible colours, but remains uncoloured,then kxw 2)( nxw k 2)( nk 2log

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.

Conjecture [Ohba] Graphs G with are chromatic choosable.

1)(2|)(| GGV

Conjecture: For any G, for any k > 1, is chromatic choosable.kG

Theorem [Noel-Reed-Wu]

On-line version of Ohba Conjecture:

Graphs G with are on-line chromatic choosable.

1)(2|)(| GGV

3 3

3 3

3 3 3

33|33|333

NOT TRUE

On-line version of Ohba Conjecture:

Graphs G with are on-line chromatic choosable.

1)(2|)(| GGV

3 3

3 3

3 3 3

33|33|333

Alice’s move

On-line version of Ohba Conjecture:

Graphs G with are on-line chromatic choosable.

1)(2|)(| GGV

3 3

3 3

3 3 3

33|33|333

Alice’s move

3|23|233

23|23|33

Bob’s (2 possible) moves

On-line version of Ohba Conjecture:

Graphs G with are on-line chromatic choosable.

1)(2|)(| GGV

3 3

3 3

3 3 3

33|33|333

Alice’s move

3|23|233

23|23|33

Bob’s (2 possible) moves

13|222

2|3|222

On-line version of Ohba Conjecture:

Graphs G with are on-line chromatic choosable.

1)(2|)(| GGV

3 3

3 3

3 3 3

33|33|333

Alice’s move

3|23|233

23|23|33

Bob’s (2 possible) moves

13|222

2|3|222

3|111

On-line version of Ohba Conjecture:

Graphs G with are on-line chromatic choosable.

1)(2|)(| GGV

3 3

3 3

3 3 3

33|33|333

Alice’s move

3|23|233

23|23|33

Bob’s (2 possible) moves

13|222

2|3|222

3|111

2|112

On-line version of Ohba Conjecture:

Graphs G with are on-line chromatic choosable.

1)(2|)(| GGV

3 3

3 3

3 3 3

33|33|333

Alice’s move

3|23|233

23|23|33

13|222

2|3|222

3|111

2|112

3|13|22

2|3|11

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

3,2 nK For n > 1, is not on-line -choosable

On-line version of Ohba Conjecture:

Graphs G with are on-line chromatic choosable.

1)(2|)(| GGV

Theorem [Erdos-Rubin-Taylor (1979)]

nK 2 is chromatic choosable.1v 2v

3v 4v

12 nv nv2

Proof

Assume each vertex is givenn permissible colours.

then colour them by a common colour use induction to colour the rest.

If for some k, have a common permissible colour

12 kv kv2 and

Assume no partite set has a common permissible colour

Build a bipartite graph

coloursBy Hall’s theorem, there is a matching that covers all the vertices of V

V

C

1v 2v 3v 4v 12 nv nv2

The proof does not work for on-line list colouring

On-line version of Ohba Conjecture: Graphs G with are on-line chromatic choosable.

1)(2|)(| GGV

Question:

nK 2 is on-line n-choosable ?

Question:

nK 2

Theorem [Huang-Wong-Z,2010]

is on-line n-choosable.

Proof

Combinatorial Nullstellensatz

An explicit winning strategy forBob ( Kim-Kwon-Liu-Z, 2012)

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

G: complete multipartite graph with partite sets

),,,,,,,(21 2121 kk BBBAAA 2|| ,1|| ii BA

NGVf )(: satisfying the following

ikf(Ai 2) 1)(

2) , 2.1)( kf(vBv i

21 2||) 2.2)( kkVf(Bi

Then (G,f) is feasible, i.e., G is on-line f-choosable.

nK 2 is on-line n-choosable.

ikf(Ai 2) 1)(

2) , 2.1)( kf(vBv i

21 2||) 2.2)( kkVf(Bi

V(G)U

Alice’s choice

UI

Bob’s choice

After this round, G changed and f changed.

Need to prove: new (G,f) still satisfies the condition

ikf(Ai 2) 1)(

2) , 2.1)( kf(vBv i

21 2||) 2.2)( kkVf(Bi V(G)U

Alice’s choice

ii BiBU colour then , somefor contains If

1by reduced 2k

1most at by reduced as holds, ) 1)( 2 )f(Aikf(A ii

2) , 2.1)( kf(vBv i

2most at by reduced as ,2||) 2.2)( 21 )f(BkkVf(B ii

ikf(Ai 2) 1)(

2) , 2.1)( kf(vBv i

21 2||) 2.2)( kkVf(Bi V(G)U

Alice’s choice

ii BiBU colour then , somefor contains If

2)( and v ,, If kvfBU uvB ii

21)(then kkuf

Bob colours v uAk 11

1by increased 1,by reduced 12 kk

ikf(Ai 2) 1)(

2) , 2.1)( kf(vBv i

21 2||) 2.2)( kkVf(Bi V(G)U

Alice’s choice

ii BiBU colour then , somefor contains If

2)( and v ,, If kvfBU uvB ii

UAj j such that index smallest theis If

jA colours Bobthen

ikf(Ai 2) 1)(

2) , 2.1)( kf(vBv i

21 2||) 2.2)( kkVf(Bi V(G)U

Alice’s choice

ii BiBU colour then , somefor contains If

2)( and v ,, If kvfBU uvB ii

UAj j such that index smallest theis If

jA colours Bobthen

v ,, If ii BU uvB

Bob colours v uA 1 1 ii AA

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

G: complete multipartite graph with partite sets

),,,,,,,(21 2121 kk BBBAAA 2|| ,1|| ii BA

NGVf )(: satisfying the following

ikf(Ai 2) 1)(

2) , 2.1)( kf(vBv i

21 2||) 2.2)( kkVf(Bi

Then (G,f) is feasible, i.e., G is on-line f-choosable.

nK 2 is on-line n-choosable.

Theorem [Kozik-Micek-Z, 2012]

G: complete multipartite graph with partite sets

),,,,,,,,,,,(321 1111 kkqk CCBBSSAA

3||,2|| ,1|| iii CBA

NGVf )(: satisfying the following

ikkf(Ai 32) 1)(

32) , 2.1)( kkf(vBv i

1||)(), 3.2)( Vvff(uCu,v i

Then (G,f) is feasible.

2or 1|| iS

||21) , )1'(1

1321

i

jji Skkkf(vSv

32) , 3.1)( kkf(vCv i

||) 2.2)( Vf(Bi

3211||) 3.3)( kkkVf(Ci

On-line version of Ohba Conjecture:

Graphs G with are on-line chromatic choosable.

1)(2|)(| GGV

Conjecture holds for graphs with independence number 3

Open Problems

Can the difference ch^{OL}-ch be arbitrarily big?

On-line Ohba conjecture true?

Nine Dragon TreeNine Dragon Tree

Thank you