2. game chromatic number

4. game chromatic number of G in the game Alice plays first the game Bob plays first game chromatic number of G in

5. Question. AliceBob 2 1 Alice ?

6. Question. AliceBob 1 1 Alice 2 Bob 2

7. Theorem. (Faigle et al.) (Kierstead and Trotter) Theorem. (Zhu) Theorem.

8. (Zhu)

9. L(p,q)-labeling

12. Question. 0 4 3 1 2 Since ? 0

13. Theorem. (Griggs and Yeh) Conjecture. (Griggs and Yeh) Theorem. (Gonçalves)

14. L(p,q)-labeling game

16. Alice plays first Bob plays first

17. Question. Alice Bob a 2 a +3 0 b Alice a b

18. Question. Alice Bob a 2 b Note.

19. Lemma.

20. Theorem. 0 2 46

21. Example. Alice 0 Bob 2 Alice plays firstBob plays first 1 3

22. Example. Alice plays first Alice 2 Bob 0 Alice 4 Bob plays first 1 3 Bob 5

23. Alice Bob 2 5 Alice 0 Bob 7 Example. Alice plays first

24. Alice Bob a 5 Alice Bob ? b

25. Bob plays first 1 Alice 5

26. Bob plays first Bob Alice 2 b Bob Alice a c

27. Question.

28. Observation 1. Alice 2 0213546 … 7 1 Bob 5 b 0

29. Observation 2. Alice 2 0213546 … 7 Bob 1 5

30. Theorem. 0213546 3 vertices, 7 numbers

31. Question. How to prove this theorem? (Use induction, we already know that

32. Idea. Alice’s strategy

33. Idea. Bob’s strategy

34. Example. 0213546798101211 13151416171918202122 2nd … 4th 1st 6th3rd 5th 7th 021345 6879101211131415 2nd … 1st 4th 3rd 5th

35. Theorem.

36. Example. 0213546 1817192120

37. Question. Alice 017 Bob

38. Bob’s strategy proof. a 27 Alice Bob 4 211016 Alice a 15

39. proof. Alice’s strategy Alice 16 Bob b Alice 28 4 15 Alice 17 Alice c 30 12 29 Bob 11 Alice

40. b 22 Bob 10 proof. Alice’s strategy 16 c 19 1 Bob 2 20 Alice

41. Theorem.

42. Thanks!

43. Note. It is not true that if Alice(resp. Bob) plays first, and at some step, he can move twice, then the smallest number needed to complete the game is less than or equal to(resp. ). For example, but if at the first step, Alice need to move twice, then the smallest number needed to complete this game is 6({0,1,2,…,6}).