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

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

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 Or there are uncolored vertices, but there is no legal color for any of the uncolored vertices Alice wins 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. 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 ] For any forest F, H.L. Bodlaender, On the complexity of some coloring games, Computer Science, 1991.

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

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

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

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, Proof: There is a forest F such that

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

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

20. 12 3

21. 12 3 4 5 6 7 8

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

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

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

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

28. Theorem [Zhu, Discrete Math.,2000] G 3k+2. ) (  g  For any partial k-tree G, Proof: 12 n kkkkk

29. Theorem [Zhu, Discrete Math.,2000] G 3k+2. ) (  g  For any partial k-tree G, Proof: 12

30. Theorem [Zhu, Discrete Math.,2000] G 3k+2. ) (  g  For any partial k-tree G, Proof: 12

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

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

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

34. Theorem [Zhu, Discrete Math.,2000] G 3k+2. ) (  g  For any partial k-tree G, 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 chromatic number game coloring number game coloring number

36. Game chromatic number AuthorGraphUpper bound Faigle, Kern, Kierstead, TrotterForests4 Guan and ZhuOuterplanar7 Faigle, Kern, Kierstead, TrotterInterval graphs3k+1 ZhuPartial k-tree3k+2 ZhuPlanar17 Game coloring number GraphLower boundUpper bound Forests44 outerplanar77 Interval graphs3k+1 Partial k-tree3k+2 Planar1117