1. On-line list colouring of graphs
Xuding Zhu
Zhejiang Normal University
CAM Hongkong

2. A scheduling problem:
There are six basketball 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

3. 1st day

4. 2nd day

5. 3rd day

6. 4th day

7. 5th day

8. This is an edge colouring problem.
Each edge is a game.
Each day is a colour.

9. A scheduling problem:
There are six basketball 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 needed
7 days are enough

10. There are 7 colours
Edge list colouring
Each edge misses at most 2 colours
Each edge has 5 permissible colours
I do not know any easy proof

11. List colouring conjecture:
For any graph G,
However, the conjecture remains open for
Haggkvist-Janssen (1997)
Uwe Schauz (2014)

12. 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
The choices are made
before the scheduling
7 days are enough

13. 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
is allowed not to show 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

14. On-line list colouring of graphs
We start colouring the graph
before having the full information of the list

15. 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
Painter
Colours vertices
Reveals the list

16. At round i
Lister choose a set of uncoloured vertices, removes
one token from each vertex of
is the set of vertices which has colour i as
a permissible colour.
Painter chooses an independent subset of
vertices in are coloured by colour i.

17. 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.

18. 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.

19. On-line list colouring:
Painter start colouring the graph
before
after having the full information of the list
choice number

20. is not 2-paintable
Theorem [Erdos-Rubin-Taylor (1979)]
is 2-choosable.

21. is not 2-paintable
Lister wins the game

22. Theorem [Erdos-Rubin-Taylor,1979]
A connected graph G is 2-choosable if and only if its core is or or
However, 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 or or

23. Problems studied
Planar graphs and locally planar graphs
Chromatic-paintable graphs
Complete bipartite graphs
Random graphs
Partial painting game
b-tuple painting game and fractional paint number
Defective painting game
Sum-painting number of graphs

24. Methods:
1. Derandomize probability arguments
2. Polynomial method
3. Inductive proof
4. Kernel method
5. Probability

25. Complete bipartite graphs
Theorem [Erdos,1964]
probabilistic proof
Theorem[Zhu,2009]
If G is bipartite and has n vertices, then

26. A B
Probability proof:
Each color is assigned to vertices in A or B with probability

27. Initially, each vertex x has weight w(x)=1B
Assume Lister has given set If
Painter colours ,
double the weight of each vertex in

28. 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
If x has permissible colours, Painter will be
able to colour it.

29. Initially, each vertex x has weight w(x)=1
Assume Lister has given set B If
Painter colours ,
double the weight of each vertex in

30. and
Theorem [RadhaKrishnan-Srinivasan,2000]
Theorem [Erdos, 1964]
Erdos-Lovasz Conjecture

31. and
Theorem [RadhaKrishnan-Srinivasan,2000]
Theorem [Erdos, 1964]
Erdos-Lovasz Conjecture
The proof uses a probability argument.
The argument CANNOT be derandomized to give a
strategy for the painting game.

32. and
Theorem [RadhaKrishnan-Srinivasan,2000]
Theorem [Erdos, 1964]
Erdos-Lovasz Conjecture
Theorem [Duray-Gutowski-Kozik,2015]
Corollary

33. Some other results proved by derandomizing probabilistic arguments
1: Partial online list colouring

34. 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.

35. Fact:
Conjecture [Albertson]:
Conjecture [Zhu, 2009]:

36. Conjecture [Zhu, 2009]:
Theorem [Wong-Zhu,2013]
Proof: Derandomize a probabilistic argument

37. Some other results proved by derandomizing probabilistic arguments
1: Partial online list colouring
2. Fractional online choice number

38. 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.

39. Theorem
[Alon-Tuza-Voigt, 1997]
[Gutowski, 2011]
Infimum attained
Infimum not attained
Probabilistic arguemnt

40. Methods:
1. Derandomize probability arguments
2. Polynomial method

41. paintable
= deg(P(G))

42. Some results proved by using polynomial method

43. Haggkvist-Janssen (1997)
Uwe Schauz (2014)

44. Methods:
1. Derandomize probability arguments
2. Polynomial method
3. Inductive proof

45. A recursive definition of f-paintable
Assume Then G is f-paintable, if
(1) or
(2)

46. Upper bounds for ch(G) proved by induction
Planar graphs
Theorem [Thomassen, 1995] Every planar graph is 5-choosable
[ Schauz,2009 ]
paintable

47. non-contractible
embedded in a surface
edge-width of G
contractible
length of shortest
non-contractible cycle
Locally planar
edge-width is large
Theorem [Thomassen, 1993] For any surface , there is a
constant , any G embedded in with edge-width >
is 5-colourable.

48. non-contractible
embedded in a surface
edge-width of G
contractible
length of shortest
non-contractible cycle
Locally planar
edge-width is large
Han-Zhu 2015
DeVos-Kawarabayashi-Mohor
2008
Theorem [Thomassen, 1993] For any surface , there is a
constant , any G embedded in with edge-width >
is 5-colourable.
choosable
paintable

49. Chromatic-paintable graphs

50. A graph G is chromatic choosable if
paintable
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
[Noel-Reed-Wu,2013]
Ohba Conjecture: Graphs G with
are chromatic choosable.
paintable

51. A graph G is chromatic choosable if
paintable
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.
Question
Ohba Conjecture: Graphs G with
are chromatic choosable.
NO!
paintable

52. Theorem [Kim-Kwon-Liu-Zhu,2012]
For k>1, is not (k+1)-paintable.

53. is not 3-paintable.

54. On-line version
Huang-Wong-Zhu 2011
Ohba Conjecture: Graphs G with
are chromatic choosable.
paintable
To prove this conjecture, we only need to consider
complete multipartite graphs.

55. 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.

56. Partition of the parts
into four classes
ordered

57. ordered
ordered

58. G is f-paintable

59. 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+]

60. d-defective painting game
At round i
Lister choose a set of uncoloured vertices, removes
one token from each vertex of
is the set of vertices which has colour i as
a permissible colour.
Painter chooses a subset of
Painter chooses an independent subset of
vertices in are coloured by colour i.

61. Questions
Theorem [ , ,1999]
No!
Planar graphs are 2-defect 3-choosable.
paintable ?
Gutowski-Han-Krawczyk-Zhu, 2016
paintable ?
Yes!
Han-Zhu, 2015
Theorem [Cushing-Kierstead,2010]
Planar graphs are 1-defect 4-choosable.
paintable ?
Open

62. Han-Zhu 2015
Locally planar graphs are 2-defective 4-paintable.

63. Methods:
1. Derandomize probability arguments
2. Polynomial method
3. Inductive proof
4. Kernel method

64. Let D be an orientation of G.
A kernel in D is an independent set I for which
every vertex not in I has an out-neighbour in I D I

65. A directed graph D is kernel perfect
if every induced sub-digraph has a kernel.
Theorem
If G has an orientation D which is kernel perfect,

66. An example:

67. Theorem [Galvin,1995] If G is bipartite, then

68. Methods:
1. Derandomize probability arguments
2. Polynomial method
3. Inductive proof
4. Kernel method
5. Probability

69. Theorem [Frieze, Mitsche,Perez-Gimenez, Pralat, 2015]

70. Theorem [Frieze, Mitsche,Perez-Gimenez, Pralat, 2015]

71. At each round, if Lister presents a large set M,
then we are sure that M contains a large independent set I.
Painter colours I.
If Lister presents a small set M, then Painter randomly colours
one vertex from the set.

72. Nine Dragon Tree
Thank you

73. Lister 33 33
333

74. Lister
Painter 3 23
233 33 33
333 23 33

75. Lister
Painter 3 23
233 33 33
333 23 33
Lister

76. Lister
Painter 3 23
233 33 33
333 23 33
Lister

77. Lister
Painter
Painter 13
222 3 23
233 33 2 3
222 33
333 23 33 3 13 22
Lister

78. Painter
Painter
Lister 13
222 3 23
233 33 2 3
222 33
333 23 33 3 13 22
Lister
Lister

79. Painter Lose
Painter
Lister
Painter
{123} 13
222 3
111 3 23
233
{1}{2}{3} 33 2 3
222 33 2
112
333 23 33
Painter Lose 3 13 22
is not 3-paintable 2 3 11
Lister
Lister
Painter Lose

80. is k-paintable
Theorem [Huang-Wong-Zhu,2011]
paintable
= deg(P(G))

81. is k-paintable
Theorem [Huang-Wong-Zhu,2011]

83. Theorem [Mahoney, Tomlinson Wise, 2014]
If G is an outerplanar graph whose weak dual is a path,
then G is online sum choice greedy.
Conjecture [Mahoney, Tolinson, Wise]
Every outerplanar graph is online sum choice greedy.