game

2. game chromatic number

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

5. Alice Alice Bob Question. 1 2 ?

6. Alice Alice Bob Bob Question. 1 1 2 2

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

8. Theorem. (Zhu)

9. L(p,q)-labeling

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

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

14. game L(p,q)-labeling

16. Alice plays first Bob plays first

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

18. Alice Bob Question. a 2 b Note.

19. Lemma. Lemma.

20. Theorem. 2 0 4 6

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

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

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

24. Alice Alice Bob Bob a b 5 ?

25. Alice Bob Bob plays first 5 1

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

27. Question.

28. Alice Bob Bob Observation 1. b 2 0 1 5 … 0 1 2 3 4 5 6 7

29. Alice Bob Observation 2. 2 5 1 … 0 1 2 3 4 5 6 7

30. Theorem. 0 1 2 3 4 5 6 3 vertices, 7 numbers

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

32. Alice’s strategy Idea.

33. Bob’s strategy Idea.

34. 4th 4th 2nd 3rd 6th 2nd 5th 3rd 1st 1st 7th 5th 0 1 2 3 4 5 … 6 7 8 9 10 11 12 13 14 15 Example. 0 1 2 3 4 5 6 7 8 9 10 11 12 … 13 14 15 16 17 18 19 20 21 22

35. Theorem.

36. Theorem. Example. 17 18 19 20 21 0 1 2 3 4 5 6

37. Bob Alice Question. 0 17

38. Alice Bob Bob Alice proof. Bob’s strategy a 27 4 a 15 10 16 21

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

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

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}).