On-line list colouring of graphs
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On-line list colouring of graphs
Xuding Zhu
Zhejiang Normal University
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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
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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
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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.
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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
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4,2,2
Theorem [Erdos-Rubin-Taylor (1979)]
n2,2,2 is 2-choosable.
4,2,2 is not on-line 2-choosable
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4,2,2
1
1
4,2,2 is not on-line 2-choosable
22 3
334
4 5
5Alice wins the game
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Question: Can the difference be arbitrarily large ?
)(ch)(chOL GG
Question: Can the ratio be arbitrarily large ?
)(ch/)(chOL GG
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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
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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).
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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
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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
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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]
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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
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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
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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
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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
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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
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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
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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
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Theorem [Kim-Kwon-Liu-Z,2012]
3,2 nK For n > 1, is not on-line -choosable
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On-line version of Ohba Conjecture:
Graphs G with are on-line chromatic choosable.
1)(2|)(| GGV
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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
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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
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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 ?
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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)
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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.
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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
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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
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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
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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
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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
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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.
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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
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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
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Open Problems
Can the difference ch^{OL}-ch be arbitrarily big?
On-line Ohba conjecture true?
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Nine Dragon TreeNine Dragon Tree
Thank you