叶鸿国 Hong-Gwa Yeh 中央大学 , 台湾 hgyeh@math.ncu.tw July 31, 2009

1. Some Results on Labeling Graphs with a Condition at Distance Two 叶鸿国Hong-Gwa Yeh 中央大学,台湾 hgyeh@math.ncu.edu.tw July 31, 2009

2. Channel-Assignment Problem

5. Hale, 1980

6. Hale, 1980, IEEE

7. 1 1

8. 1 1

9. 2 1

10. 1 2 2 2 3 1 3 1 1 3

11. 1 Chromatic number = 3 2 2 2 3 1 3 1 1 3

12. However, interference phenomena may be so powerful that even the different channels used at “very close” transmitters may interfere.

13. Roberts, 1988 ? “close” transmitters must receive different channels and “very close”transmitters must receive channels that are at least two channels apart. ?

14. Griggs and Yeh, 1992, SIAM J. Discrete Math. k-L(2,1)-labeling of a graph G

15. k-L(2,1)-labeling of a graph G f:V(G)-------->{0,1,2,…,k} s.t. |f(x)-f(y)|≧2 if d(x,y)=1 |f(x)-f(y)|≧1 if d(x,y)=2 J. R. GRIGGS R. K. YEH

16. 1 Roberts, 1980 2 2 2 3 1 3 1 1 3

17. 8-L(2,1)-labeling of P 7-L(2,1)-labeling of P 6-L(2,1)-labeling of P ? ? ?

18. 8 3 9-L(2,1)-labeling of P ?

19. 8 3 9-L(2,1)-labeling of P ? λ(G) =λ-number of G λ(P)=9

20. The problem of determining λ(G) for general graphsG is known to be NP-complete!

21. Goodupper boundsfor λ(G) are clearly welcome.

22. Griggs and Yeh: λ(G) ≦△2+2△ Chang and Kuo:λ(G) ≦△2+△ Kral and Skrekovski : λ(G) ≦△2+△-1 Goncalves:λ(G) ≦△2+ △-2

23. Griggs-Yeh Conjecture 1992 J. R. GRIGGS λ(G) ≦△2 for any graph G with maximum degree △≧2 R. K. YEH

24. Very recently Havet, Reed, and Sereni have shown that Griggs-Yeh Conjecture holds for sufficiently large△ !! SODA 2008

25. Note that to prove Griggs-Yeh Conjecture it suffices to consider regular graphs.

26. However….

27. Very little was known about exact L(2,1)-labeling numbers for specific classes of graphs. --- even for3-regular graphs

28. Consider various subclasses of 3-regular graphs Kang, 2008, SIAM J. on Discrete Math., proved that Griggs-Yeh Conjecture is true for 3-regular Hamiltoniangraphs

29. Other important subclasses of 3-regular graphs Generalized Petersen Graph

30. GeneralizedPetersenGraph of order 5 GPG(5)

31. GPG(3) , GPG(4)

32. GPG(6)

33. GPG(9)

34. Griggs-Yeh Conjecture says that λ(G) ≦9for all GPGs G

35. Georges and Mauro, 2002, Discrete Math. λ(G) ≦8for all GPGs G except for the Petersen graph

36. Georges and Mauro, 2002, Discrete Math. λ(G) ≦7for all GPGs G of order n≦6 except for the Petersen graph

37. Georges-Mauro Conjecture 2002 For any GPG G of order n≧7, λ(G) ≦7

38. Jonathan Cass Denise Sakai Troxell Sarah Spence Adams 2006, IEEE Trans. Circuits & Systems Georges-Mauro Conjecture is true for orders 7 and 8

39. More….

40. Number of non-isomorphic GPGs of order n with the aid of a computer program

41. Y-Z Huang, C-Y Chiang, L-H Huang, H-G Yeh 2009 Georges-Mauro Conjecture is true for orders 9,10,11 and 12

42. Generalized Petersen graphs of orders 9, 10, 11 and 12 Theorem One-page proof !! 42

43. 3 3 3 43

44. 3 1, 2, 4, 5, 6 3 3 44

45. Case 7 Case 3 Case 5 Case 6 Case 2 Case 1 Case 4 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 45

46. Case 1 3 5 1 6 2 0 1, 2, 4, 5, 6 4 4 7 2 3 1 3 5 46

47. Case 1 3 Case A 5 1 6 2 0 1, 2, 4, 5, 6 4 4 7 2 3 1 3 5 47

48. Case 1 3 Case A 0 7 5 1 6 2 7 0 0 1, 2, 4, 5, 6 4 4 7 2 3 1 3 5 0 7 48

49. Case 1 3 Case B 5 1 6 2 0 1, 2, 4, 5, 6 4 4 7 2 3 1 3 5 49

50. Case 1 3 Case B 7 0 5 1 6 2 0 7 0 1, 2, 4, 5, 6 4 4 7 2 3 1 3 5 7 0 50