1. Some Results on Labeling Graphs with a Condition at Distance Two
叶鸿国 Hong-Gwa Yeh
中央大学,台湾
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. interference phenomena may be so powerful
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. k-L(2,1)-labeling of a graph G
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. determining λ(G) for general graphs G is known to be
The problem of
determining λ(G) for general graphs G is known to be
NP-complete!

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

25. to prove Griggs-Yeh Conjecture to consider regular graphs.
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 for 3-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 Hamiltonian graphs

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

30. Generalized Petersen Graph of order 5
GPG(5)

31. GPG(3) , GPG(4)

32. GPG(6)

33. GPG(9)

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

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

36. Georges and Mauro, 2002,
Discrete Math.
λ(G) ≦7 for 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. IEEE Trans. Circuits & Systems Georges-Mauro Conjecture
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. non-isomorphic GPGs of order n
Number of
non-isomorphic GPGs of order n
with the aid of
a computer program

41. Georges-Mauro Conjecture
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 42

43. 3 3 3 43 43

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

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

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

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

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

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

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

51. Case 2 3 5 1 2 4
1, 2, 4, 5, 6 6 4 7 2 3 1 3 5 51 51

52. Case 2 3
Case A 7 5 1 2 4 7
1, 2, 4, 5, 6 6 4 7 2 3 1 3 5 7 52 52

53. Case 2 3
Case B 7 5 1 2 4 7
1, 2, 4, 5, 6 6 4 7 2 3 1 3 5 7 53 53

54. Case 7 3 7 6 4 2 2 7
1, 2, 4, 5, 6 6 4 7 4 3 6 3 2 7 54 54

55. Theorem 55

56. 56

57. 1 Case 1 1 Case 2 1 Case 3 1 Case 4 1 Case 5 1 Case 6 1 Case 7 Case 857

58. 1
4, 4, 6 1 1 58

59. Case 1 1 4 6 6
0, 2, 4, 6 5 7 3 4 2 1 4 2 1 59

60. Case 8
再次一个
从这开始
次一个 60

61. Case 8 3 6 1 6 7 6 1 3 4 5 4 4 2 7 2 1 5 61

62. 太过暴力, 不宜在此陈述! .
其余的证明呢?

64. 64