Painting game on graphs
Xuding Zhu
Zhejiang Normal University
2014.05.28
8th Shanghai Conference on Combinatorics
A scheduling problem:
There are six teams, each needs to compete
with all the others.
Each team can play one game per day
How many days are needed to schedule
all the games?
Answer: 5 days
1st day
2nd day
3rd day
4th day
5th day
This is an edge colouring problem.
Each day is a colour.
 ' ( K6 )  5
 ' ( K 2 n )  2n  1
A scheduling problem:
There are six teams, each needs to compete
with all the others.
Each team can play one game per day
Each team can choose one day off
How many days are needed to schedule
all the games?
Answer: 5 days
7 days are enough
7 days are needed
There are 7 colours
Each edge misses at most 2 colours
ch' ( K 6 )  5
Each edge has 5 permissible colours
I do not know any easy proof
List colouring conjecture:
For any graph G,
ch ' (G )   ' (G )
However, the conjecture remains open for K 2n
Haggkvist-Janssen (1997)
ch ' ( K n )  n
2n- 1  '(K 2n )  ch'(K 2n )  2n
A scheduling problem
There are six teams, each needs to compete
with all the others.
Each team can play one game per day
Each team can choose one day off
How many days are needed to schedule
all the games?
Answer: 5 days
7 days are enough
The choices are made
before the scheduling
A scheduling problem
There are six teams, each needs to compete
with all the others.
Each team can play one game per day
is allowed
not day
to show
Each team can
choose one
off up for one day
How many days are needed to schedule
all the games?
On each day, we know
which teams haven’t shown up today
7 days are enough
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 : V (G)  0,1,2,
f (x) is the number of permissible colours for x
f-painting game (on-line list colouring game) on G
Each vertex v is given f(v) tokens.
Each token represents a permissible colour.
But we do not know yet what is the colour.
Two Players:
Lister
Reveal the list
Painter
Colour vertices
At round i
Lister choose a set Vi of uncoloured vertices, removes
one token from each vertex of Vi
Vi is the set of vertices which has colour i as
a permissible colour.
Painter chooses an independent subset I i of Vi
vertices in I i are coloured by colour i.
If at the end of some round, there is an uncolored
vertex with no tokens left, then Lister wins.
If all vertices are coloured then
Painter 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 ch p(G ) of G is the
minimum k for which G is k-paintable.
On-line
List colouring:
list colouring:
Painter start colouring the graph
after having the full information of the list
before
choice number
ch(G)  min k : G is k - choosable 
ch p(G )  ch(G )
 2, 2, 4
is not 2-paintable
 2, 2, 4
Theorem [Erdos-Rubin-Taylor (1979)]
 2, 2, 2 n
is 2-choosable.
 2, 2, 4
is not 2-paintable
 2, 2, 4
Lister wins the game
3
4
5
3
5
1
4
2
3
2
1
Theorem [Erdos-Rubin-Taylor,1979]
A connected graph G is 2-choosable if and only if its core is
K1
or
C 2n
or
 2,2,2 p
However, if p>1, then  2,2,2 p is not 2-paintable.
Theorem [Zhu,2009]
A connected graph G is 2-paintable if and only if its core is
K1
or
C 2n
or
 2,2,2
A recursive definition of f-paintable
Assume f : V (G)  0,1,2, . Then G is f-paintable, if
(1) V (G )  
(2)
or
X  V (G ),  independent set I
G-I is (f-δX )  paintable.
 X : characteristic function of X
 X,
For any question about list colouring,
we can ask the same question for on-line list colouring
For any result about list colouring,
we can ask whether it holds for on-line list colouring
Planar graphs and locally planar graphs
Chromatic-paintable graphs
Complete bipartite graphs
Partial painting game
b-tuple painting game and fractional paint number
Some upper bounds for ch(G ) are automatically
upper bounds for ch p(G )
Upper bounds for ch(G) proved by kernel method are also upper
bounds for ch p(G )
Theorem [Galvin,1995] If G is bipartite, then  'p (G )  (G )
Upper bounds for ch(G) proved by Combinatorial Nullstellensatz
are also upper bounds for ch p(G )
Theorem [Schauz, 2009]
If G has an orientation D for which
|EE(D)|  |OE(D)| ,
then G is (d D  1)  paintable
Brooks’ Theorem [Hladky-Kral-Schauz,2010]
paintable
If G  K n , C 2k 1 , then G is   choosable
Upper bounds for ch(G) proved by induction
Planar graphs
[ Schauz,2009
] Every planar graph is 5-choosable
Theorem [Thomassen,
1995]
paintable
Thomassen proves a stronger result:
Assume G is a planar graph with boundary cycle
C  (v 1 ,v 2 , ,v k )
If f(v1 )  f(v 2 )  1, f(v i )  3 for 3  i  k ,
and f(v )  5 for v  V (G )  C ,
paintable
then G is f - choosable.
Basically, Thomassen’s proof works for f-paintable.
G embedded in a surface 
non-contractible
edge-width of G
length of shortest
non-contractible cycle
contractible
Locally planar
edge-width is large
： torus
Theorem [Thomassen, 1993] For any surface  , there is a
constant w , any G embedded in  with edge-width > w
is 5-colourable.
G embedded in a surface 
non-contractible
edge-width of G
length of shortest
non-contractible cycle
contractible
Locally planar
edge-width is large
： torus
Han-Zhu
DeVos-Kawarabayashi-Mohor
2014+
2008
Theorem [Thomassen,
1993] For any surface  , there is a
constant w , any G embedded in  with edge-width > w
is 5-colourable.
choosable
paintable
Find a subgraph H
Apply strategy for planar graphs on the pieces
of H and on G-H, one by one
G-H is planar
Each piece in H is planar
+ some other nice properties
Chromatic-paintable graphs
A graph G is chromatic choosable
paintable if ch
chp((GG)) (G(G) )
Conjecture: Line graphs are chromatic choosable.
paintable
Conjecture: Claw-free graphs are chromatic choosable.
paintable
Conjecture: Total graphs are chromatic choosable.
paintable
[Kim-Park,2013]
Conjecture: Graph squares are chromatic choosable.
Theorem
Ohba Conjecture: Graphs G with | V (G ) | 2  (G )  1
[Noel-Reed-Wu,2013] are chromatic choosable.
paintable
A graph G is chromatic choosable
paintable if ch
chp((GG)) (G(G) )
Conjecture: Line graphs are chromatic choosable.
paintable
Conjecture: Claw-free graphs are chromatic choosable.
paintable
Conjecture: Total graphs are chromatic choosable.
paintable
[Kim-Park,2013]
Conjecture: Graph squares are chromatic choosable.
Ohba
Conjecture: Graphs G with | V (G ) | 2  (G )  1 NO!
Question
are chromatic choosable.
paintable
K 2, 2,3 is not 3-paintable.
Lister
33
33
333
Lister
33
33
333
Painter
3
23
233
23
23
33
Lister
33
33
333
Painter
3
23
233
23
23
33
Lister
Lister
33
33
333
Painter
3
23
233
23
23
33
Lister
Lister
33
33
333
Painter
3
23
233
23
23
33
Lister
Painter
13
222
2
3
222
3
13
22
Lister
33
33
333
Painter
3
23
233
23
23
33
Lister
Painter
13
222
2
3
222
3
13
22
Lister
Lister
33
33
333
K 2, 2,3
is not 3-paintable
Painter
3
23
233
23
23
33
Lister
Painter
13
222
2
3
222
3
13
22
Lister
Painter Lose
3 {123}
111
{1}{2}{3}
2
112
Painter Lose
2
3
11
Painter Lose
Theorem [Kim-Kwon-Liu-Zhu,2012]
For k>1,
K 2k ,3 is not (k+1)-paintable.
On-line version
Huang-Wong-Zhu 2011
Ohba Conjecture: Graphs G with | V (G ) | 2  (G )  1
are chromatic choosable.
paintable
To prove this conjecture, we only need to consider
complete multipartite graphs.
Theorem [Huang-Wong-Zhu,2011]
K 2n
is n-paintable
v1
v2
v3
v4
v2 n 1
v2 n
The first proof is by using
Combinatorial 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 graphs
with independence number at most 3.
The key in proving this theorem is to find a “good”
technical statement that can be proved by induction.
ordered
parts of size 1
parts of size 2
parts of size 3
parts of size 1 or 2
ordered
Partition of the parts
into four classes
A1
A2
ordered
parts of size 1
parts of size 2
parts of size 3
A k1
k2
k3
Bi
Ci
S1
parts of size 1 or 2
ordered
S2
Ss
vS (i ) 
| S
1 j  i
j
|
For Ai  v
f (v )  k 2  k 3  i
For Bi  u, v
f (v )  k 2  k 3
G is f-paintable
f (v)  f (u ) | V (G ) |
For Ci  u, v, w
f (v )  k 2  k 3
f (v)  f (u ) | V (G ) | 1
f (v)  f (u)  f (w) | V (G) | 1  k1  k2  k3
For v  Si
f (v)  k1  k2  2k3  vS (i)
Theorem [Kozik-Micek-Zhu,2014]
On-line Ohba conjecture is true for graphs
with independence number at most 3.
Theorem [Chang-Chen-Guo-Huang,2014+]
m2  m  2
If (G )  m , | V | 2
(G )
m  3m  4
then G is chromatic - paintable
(g )  4,| V |
7
(G )    paintable
4
Complete bipartite graphs
Theorem [Erdos,1964]
If G is bipartite and has n vertices , then
ch(G)  log 2 n  1
Theorem[Zhu,2009]
If G is bipartite and has n vertices, then
chp(G )  log2 n  1
probabilistic proof
Initially, each vertex x has weight w(x)=1
A
Assume Lister has given set
If
Vi
B
w( A  Vi )  w( B  Vi )
Painter colours A  Vi ,
double the weight of each vertex in
B  Vi
A
The total weight of uncoloured vertices
is not increased.
B
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
ww((xx))  22kk  n
k  log 2 n
If x has log 2 n  1 permissible colours, Painter will be
able to colour it.
Theorem [Carraher-Loeb-Mahoney-Puleo-Tsai-West]
t1
K k ,r
t2
f : V  1,2,...,
tk
X
k
k
K k ,r is f - paintable
iff r 
k
ti

i
1
k Y
k | X || Y | r
3-choosable complete bipartite graphs
Theorem
K p ,q is 3 - choosable (p  q) iff
p  3,q  26
p  4,q  18
Mahadev-Roberts
-Santhanakrishnan, 1991
p  5,q  8
Furedi-Shende-Tesman, 1995
p  6,q  7
O-Donnel,1997
3-paintable complete bipartite graphs
Theorem [Chang-Zhu,2013]
K p ,q is 3 - paintable (p  q) iff
p  3,q  26
p  4,q  11
p  5,q  8
p  6,q  7
ch(K n ,n ) and chp(K n ,n )
Theorem [Erdos,
1964]
Erdos-Lovasz
Conjecture
ch(K n ,n )ch
(Klog
n (1 (1 o(1o))
(1))
log
log
n ,n )2 
2 n2 log 2 n
1
log2 n  (2  o(1))log2 log2 n  ch(K n ,n )  log2 n  (  o(1))log2 log2 n
2
Maybe
Theorem [Zhu, 2010]
(K )  (1  o(1))log 2 n 2 n ?
chp(Kch
n ,np) n ,nlog 2 n  o(1) log 2 log
log 2 n  log 2 log 2 n  log 2 log 2 log 2 n  ch p(K n ,n )  log 2 n  2
ch(K n ,n ) and chp(K n ,n )
Theorem [RadhaKrishnan-Srinivasan,2000]
1

ch(K n ,n )  log2 n    o(1) log2 log2 n
2

Probabilistic mothed.
Theorem [Duray-Gutowksi-Kozik,2014+]
chp(K n ,n )  log2 n  log 2 log 2 n
Theorem [Gerbner-Vizer, 2014+]
chp(K n ,n )  log2 n  log2 log 2 n  log2 log 2 log 2 n
chp(K n ,n )  log 2 n  c log 2 log 2 n for some constant c  1
?
b-tuple list colouring
G is (a,b)-choosable
if |L(v)|=a for each vertex v, then there is a b-tuple
L-colouring.
b-tuple on-line list colouring
If each vertex has a tokens, then Painter has a
strategy to colour each vertex a set of b colours.
Conjecture [Erdos-Rubin-Taylor]
(a,b)-choosable
(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
a

f (G )  inf : G is (a,b) - colourable
b

a

chf (G )  inf : G is (a,b) - choosable
b

a

chfp(G )  inf : G is (a,b) - paintable
b

Theorem [Alon-Tuza-Voigt, 1997]
f (G )  chf (G )
Infimum attained
[Gutowski, 2011]
f (G )  chfP(G )
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
K1
or
C 2n
or
 2,2,2
Theorem [Alon-Tuza-Voigt, 1997]
f (G )  chf (G )
Infimum attained
[Gutowski, 2011]
f (G )  chfP(G )
Infimum not attained
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:
If (G )  k , then using k' k colours, one can
colour at least
k'
V
k
vertices.
Conjecture [Albertson]:
If ch(G )  k , each vertex has k' k permissible
colours, one can colour at least
k'
V
k
vertices.
Conjecture [Zhu, 2009]:
If ch P(G )  k , each vertex has k' k tokens
then Painter has a strategy to colour at least
k'
V
k
vertices.
Conjecture
Theorem
[Wong-Zhu,2013]
[Zhu, 2009]:
If ch P(G )  k , each vertex has k' k tokens
then Painter has a strategy to colour at least
6 k'
vertices.
V
7 k
Question: Can the difference
chp(G )  ch(G )
be arbitrarily large ?
Nine Dragon Tree
Thank you
An easy but useful lemma:
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.
Corollary: If G is k-degenerate, then G is (k+1)-paintable
vk
v1
v2
u
Case 1
w
vk
v1
v2
u ,w play the role
of v 1 ,v 2 in G 2
u
G1
Vi
G2
I i  I i1  I i2
Case 1
w
Painter apply his winning strategy on G 1 with
Vi1  Vi  V (G 1 ) , obtain independent set I i1  Vi1
Then Painter apply his winning strategy on G 2 with


Vi2  I i1  u ,w   Vi  V (G 2 )  u ,w 
obtain independent set I i2  Vi2
vk
v1
v2
Vi
Case 2
Vi '
defined as follows:
If v 1  Vi , then Vi ' Vi  v k 
If v 1  Vi and v k  Vi , then Vi ' Vi  N G [v k ]
Otherwise Vi ' Vi
Painter apply his winning strategy on G  v k with input Vi '
obtains I i ' add v k if possible.