叶鸿国 Hong-Gwa Yeh中央大学 , 台湾 [email protected] 31, 2009
Some Results on Labeling Graphs with a Condition at Distance Two
2
Channel-Assignment Problem
6
Hale, 1980, IEEE
10
2
1
3
3 1
1
2
3
2
1
11
2
1
3
3 1
1
2
3
2
1 Chromatic number = 3
12
However, interference phenomena may be
so powerful that even the different channels
used at “very close” transmitters may interfere.
13
“close” transmitters must receive different channels
and “very close” transmitters must
receive channels that are at least two channels apart.
Roberts, 1988
?
?
14
k-L(2,1)-labeling of a graph G
Griggs and Yeh, 1992, SIAM J. Discrete Math.
15
J. R. GRIGGS
R. K. YEH
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
k-L(2,1)-labeling of a graph G
16
2
1
3
3 1
1
2
3
2
1 Roberts, 1980
17
8-L(2,1)-labeling of P
7-L(2,1)-labeling of P
6-L(2,1)-labeling of P ?
??
18
9-L(2,1)-labeling of P
83
?
19
9-L(2,1)-labeling of P
λ(G) =λ-number of Gλ(P)=9
83
?
20
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
J. R. GRIGGS
R. K. YEH
Griggs-Yeh Conjecture1992
λ(G) ≦△2 for any graph G with maximum degree 2△≧
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.
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)
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 Conjecture2002
For any GPG G of order n 7,≧
λ(G) ≦7
38
Sarah Spence Adams
Jonathan Cass
Denise Sakai Troxell
2006, IEEE Trans. Circuits & Systems
Georges-Mauro Conjectureis true
for orders 7 and 8
39
Generalized Petersen6
grap of order h 6
More….
40
Number of non-isomorphic GPGs of order n
with the aid of a computer program
41
Georges-Mauro Conjectureis true
for orders 9,10,11 and 12
Y-Z Huang, C-Y Chiang,L-H Huang, H-G Yeh
2009
42
Theorem
Generalized Petersen7
gra of order 9ph
Generalized Petersen graphs of orders 9, 10, 11 and 12
One-page proof !!
44
3
33
1, 2, 4, 5, 6
45
3
33
Case 1 3
33
Case 2 3
33
Case 3
3
33
Case 4 3
33
Case 5 3
33
Case 6
3
33
Case 7
46
3
33
5 1
6
4
25
1
4
2
1, 2, 4, 5, 60
7
Case 1
47
3
33
5 1
6
4
25
1
4
2
1, 2, 4, 5, 60
7
Case 1Case A
48
3
33
0
7
07
0
75 1
6
4
25
1
4
2
1, 2, 4, 5, 60
7
Case 1Case A
49
3
33
5 1
6
4
25
1
4
2
1, 2, 4, 5, 60
7
Case 1Case B
50
3
33
7
0
70
7
05 1
6
4
25
1
4
2
1, 2, 4, 5, 60
7
Case 1Case B
51
3
33
5 1
6
25
1
4
2
1, 2, 4, 5, 60
7
Case 2
4
52
3
33
7
0
70
7
05 1
6
25
1
4
2
1, 2, 4, 5, 60
7
Case 2
4
Case A
53
3
33
0
7
07
0
75 1
6
25
1
4
2
1, 2, 4, 5, 60
7
Case 2
4
Case B
54
3
33
0
7
07
0
76
6
46
2
2
4
2
1, 2, 4, 5, 60
7
Case 7
4
55
Theorem
Generalized Petersen7
grap of order h 10
57
1
1
1
Case 1 1
1
1
Case 2 1
1
1
Case 3
1
1
1
Case 4 1
1
1
Case 5 1
1
1
Case 6
1
1
1
Case 7 Case 8
59
1
1
1
242
0
0
06
Case 1
46
4
0, 2, 4, 65
7
3
60
Case 8
从这开始
次一个
再次一个
61
3
44
6
1
51
6
0
5
7
4
0
7
2
0
2
1
3
6
Case 8
太过暴力 , 不宜在此陈述 ! .其余的证
明呢 ?