1. Game chromatic number of graphs
吳佼佼
中研院數學所

2. Two players: Alice and Bob
A graph G
A set of colors

3. Adjacent vertices cannot be colored
by the same color.

4. Game over !

5. Game is over when one cannot make a move.
Either all the vertices are colored
Alice wins
Or there are uncolored vertices, but there is
no legal color for any of the uncolored
vertices
Bob wins

6. Alice’s goal: have all the vertices colored.
Bob’s goal: to have an uncolored vertex with no legal color.

7. In the previous example, Bob wins the game !
But, Alice could have won the game if she had
played carefully !
If both players play “perfectly”, who will win the game ?

8. It depends on the graph G, and depends on the number of colors !
Given a graph G, the game chromatic number of G
is the least number of colors for which
Alice has a winning strategy.

9. A hint that the problem is difficult.
To prove that
one needs to prove the correctness of a sentence of
the form:
MA: a move for Alice;
MB: a move for Bob
A hint that the problem is difficult.

10. Theorem [Faigle,Kern,Kierstead,Trotter, Ars. Combin., 1993 ]
H.L. Bodlaender, On the complexity of some coloring games, Computer Science, 1991.
Theorem [Faigle,Kern,Kierstead,Trotter, Ars. Combin., 1993 ]
For any forest F,

11. 7. ) ( £ c Theorem [Guan and Zhu, J. Graph Theory, 1999]
For any outerplanar graph G, G 7. ) ( g c

12. 3k+2. ) ( £ c Theorem [Zhu, Discrete Math.,2000]G
3k+2. ) ( g c
For any partial k-tree G,
2-tree
A partial k-tree is a subgraph of a k-tree

13. Theorem [Zhu]
For any planar graph G, G
17. ) ( g c

14. Theorem [Faigle,Kern,Kierstead,Trotter, Ars. Combin., 1993 ]
For any forest F,
Proof:

15. Theorem [Faigle,Kern,Kierstead,Trotter, Ars. Combin., 1993 ]
For any forest F,
Proof:

16. Theorem [Faigle,Kern,Kierstead,Trotter, Ars. Combin., 1993 ]
For any forest F,
Proof:

17. Theorem [Faigle,Kern,Kierstead,Trotter, Ars. Combin., 1993 ]
For any forest F,
There is a forest F such that
Proof:

18. 7. ) ( £ c G’ Theorem [Guan and Zhu, J. Graph Theory, 1999]
For any outerplanar graph G, G 7. ) ( g c
Proof: G’

19. Theorem [Guan and Zhu, J. Graph Theory, 1999]
For any outerplanar graph G, G 7. ) ( g c
Proof: 7 2 10 12 G’ 1 11 3 9 4 8 5 6

20. 1 2 3

21. 1 2 4 3 5 6 7 8

22. 1 2 3 4 5 6 7 8 9 10 11 12 14 13 15 16 17 18 19 20 21 22 23 24

23. For each uncolored vertex v,
there are at most 3 colored neighbours in T.

24. 1 2 3 4 5 For each uncolored vertex v, there are at most
6 colored neighbours in G’. 1 2 3 4 5 6 7 8 9 10 11 12 14 13 15 16 17 18 19 20 21 22 23 24

25. 1 2 3 4 5 For each uncolored vertex v, there are at most
6 colored neighbours in G. 1 2 3 4 5 6 7 8 9 10 11 12 14 13 15 16 17 18 19 20 21 22 23 24

26. 7. ) ( £ c Theorem [Guan and Zhu, J. Graph Theory, 1999]
For any outerplanar graph G, G 7. ) ( g c
There is an outerplanar G such that

27. 3k+2. ) ( £ c Theorem [Zhu, Discrete Math.,2000]
For any partial k-tree G, G
3k+2. ) ( g c
Proof: 1 2 n
k+1
k+2
k+3 k

28. 3k+2. ) ( £ c Theorem [Zhu, Discrete Math.,2000]
For any partial k-tree G, G
3k+2. ) ( g c
Proof: k k k k k 1 2 n

29. 3k+2. ) ( £ c Theorem [Zhu, Discrete Math.,2000]
For any partial k-tree G, G
3k+2. ) ( g c
Proof: 1 2

30. 3k+2. ) ( £ c Theorem [Zhu, Discrete Math.,2000]
For any partial k-tree G, G
3k+2. ) ( g c
Proof: 1 2

31. 3k+2. ) ( £ c Theorem [Zhu, Discrete Math.,2000]
For any partial k-tree G, G
3k+2. ) ( g c
Proof:
Uncolored vertex x

32. 3k+2. ) ( £ c Theorem [Zhu, Discrete Math.,2000]
For any partial k-tree G, G
3k+2. ) ( g c
Proof:
Uncolored vertex x

33. 3k+2. ) ( £ c Theorem [Zhu, Discrete Math.,2000]
For any partial k-tree G, G
3k+2. ) ( g c
Proof:
Uncolored vertex x

34. There are at most k +2(k+1) colored neighbours of x.
Theorem [Zhu, Discrete Math.,2000]
For any partial k-tree G, G
3k+2. ) ( g c
Proof:
Uncolored vertex x
There are at most k +2(k+1) colored neighbours of x.
k +2k+1=3k+1

35. t colors
game coloring number
game chromatic number game coloring number

36. Game chromatic number
Author
Graph
Upper bound
Faigle, Kern, Kierstead, Trotter
Forests 4
Guan and Zhu
Outerplanar 7
Interval graphs
3k+1
Zhu
Partial k-tree
3k+2
Planar 17
Game coloring number
Graph
Lower bound
Upper bound
Forests 4
outerplanar 7
Interval graphs
3k+1
Partial k-tree
3k+2
Planar 11 17

37. The End