Some Results on
Labeling Graphs with
a Condition at Distance Two
叶鸿国 Hong-Gwa Yeh
中央大学,台湾
[email protected]
July 31, 2009
Channel-Assignment Problem
2
3
4
Hale, 1980
5
Hale, 1980, IEEE
6
1
1
7
1
1
8
2
1
9
1
2
2
2
3
1
3
1
1
3
10
1
Chromatic number = 3
2
2
2
3
1
3
1
1
3
11
However,
interference phenomena may be
so powerful
that even the different channels
used at “very close” transmitters
may interfere.
12
Roberts, 1988
“close” transmitters must
?receive
different channels
and
“very close” transmitters must
receive channels that are
at least two channels apart.
?
13
Griggs and Yeh,
1992, SIAM J. Discrete Math.
k-L(2,1)-labeling of a graph G
14
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
15
1
Roberts, 1980
2
2
2
3
1
3
1
1
3
16
8-L(2,1)-labeling of P ?
7-L(2,1)-labeling of P ?
6-L(2,1)-labeling of P ?
17
3
8
9-L(2,1)-labeling of P
?
18
3
8
9-L(2,1)-labeling of P
?
λ(G) =λ-number of G
λ(P)=9
19
The problem of
determining λ(G) for general graphs G is
known to be
NP-complete!
20
Good upper bounds for λ(G)
are clearly welcome.
21
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
22
Griggs-Yeh Conjecture
1992
J. R. GRIGGS
λ(G) ≦△2
for any graph G
with maximum degree △≧2
R. K. YEH
23
Very recently
Havet, Reed, and Sereni
have shown that
Griggs-Yeh Conjecture holds
for sufficiently large △ !!
SODA 2008
24
Note that
to prove Griggs-Yeh Conjecture
it suffices
to consider regular graphs.
25
However….
26
Very little was known about
exact L(2,1)-labeling numbers for
specific classes of graphs.
--- even for 3-regular graphs
27
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
28
Other important
subclasses of 3-regular graphs
Generalized Petersen Graph
29
Generalized Petersen Graph of order 5
GPG(5)
30
GPG(3) , GPG(4)
31
GPG(6)
32
GPG(9)
33
Griggs-Yeh Conjecture
says that
9
λ(G) ≦ for all GPGs G
34
Georges and Mauro, 2002,
Discrete Math.
8
λ(G) ≦ for all GPGs G
except for
the Petersen graph
35
Georges and Mauro, 2002,
Discrete Math.
7
λ(G) ≦ for all GPGs G
of order n≦6
except for
the Petersen graph
36
Georges-Mauro Conjecture
2002
For any GPG G of order n≧7,
7
λ(G) ≦
37
Jonathan Cass
Sarah Spence Adams
Denise Sakai Troxell
2006,
IEEE Trans. Circuits & Systems
Georges-Mauro Conjecture
is true
for orders 7 and 8
38
More….
 Generalized Petersen 

6
 graph of order  6 
39
Number of
non-isomorphic GPGs of order n
with the aid of
a computer program
40
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
41
Generalized Petersen graphs
of orders 9, 10, 11 and 12
Theorem
 Generalized Petersen 

7
 graph of order  9 
One-page proof !!
42
3
3
3
43
3
1, 2, 4, 5, 6
3
3
44
Case 1
3
3
Case 4
3
3
3
Case 3
3
3
3
Case 5
3
3
Case 7
Case 2
3
3
3
3
Case 6
3
3
3
3
3
3
3
45
Case 1
3
5
1
2
6
0
4
3
1, 2, 4, 5, 6
4
7
1
2
3
5
46
Case 1
3
Case A
5
1
2
6
0
4
3
1, 2, 4, 5, 6
4
7
1
2
3
5
47
Case 1
Case A
3
0
7
5
1
2
6
0
7
0
4
3
1, 2, 4, 5, 6
4
7
1
7
2
3
5
0
48
Case 1
3
Case B
5
1
2
6
0
4
3
1, 2, 4, 5, 6
4
7
1
2
3
5
49
Case 1
Case B
3
7
0
5
1
2
6
7
0
0
4
3
1, 2, 4, 5, 6
4
7
1
0
2
3
5
7
50
Case 2
3
5
1
2
4
0
4
3
1, 2, 4, 5, 6
6
7
1
2
3
5
51
Case 2
Case A
3
7
0
5
1
2
4
7
0
0
4
3
1, 2, 4, 5, 6
6
7
1
0
2
3
5
7
52
Case 2
Case B
3
0
7
5
1
2
4
0
7
0
4
3
1, 2, 4, 5, 6
6
7
1
7
2
3
5
0
53
Case 7
3
0
7
4
6
2
0
2
0
4
3
7
1, 2, 4, 5, 6
6
7
6
7
4
3
2
0
54
Theorem
 Generalized Petersen 

7
 graph of order  10 
55
56
Case 1
Case 2
1
Case 5
1
Case 6
1
1
1
1
1
1
Case 7
1
1
1
1
1
1
1
Case 4
Case 3
1
1
1
Case 8
1
1
1
57
1
4, 4, 6
1
1
58
Case 1
1
4
0
6
6
0
3
5
0, 2, 4, 6
7
2
4
0
4
1
2
1
59
Case 8
再次一个
从这开始
次一个
60
Case 8
3
0
6
7
6
4
1
6
1
0
3
4
5
2
0
4
7
2
1
5
61
其余的证
明呢?
太过暴
力, 不宜
在此陈
述! .
63
64