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

Some Results on Labeling Graphs with a Condition at Distance Two

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Channel-Assignment Problem

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4

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Hale, 1980

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Hale, 1980, IEEE

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11

8

11

9

12

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

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However, interference phenomena may be

so powerful that even the different channels

used at “very close” transmitters may interfere.

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“close” transmitters must receive different channels

and “very close” transmitters must

receive channels that are at least two channels apart.

Roberts, 1988

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k-L(2,1)-labeling of a graph G

Griggs and Yeh, 1992, SIAM J. Discrete Math.

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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

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2

1

3

3 1

1

2

3

2

1 Roberts, 1980

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8-L(2,1)-labeling of P

7-L(2,1)-labeling of P

6-L(2,1)-labeling of P ?

??

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9-L(2,1)-labeling of P

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9-L(2,1)-labeling of P

λ(G) =λ-number of Gλ(P)=9

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The problem of

determining λ(G) for general graphs G is known to be

NP-complete!

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Good upper bounds for λ(G) are clearly welcome.

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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△

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J. R. GRIGGS

R. K. YEH

Griggs-Yeh Conjecture1992

λ(G) ≦△2 for any graph G with maximum degree 2△≧

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Very recently Havet, Reed, and Sereni

have shown that Griggs-Yeh Conjecture holds

for sufficiently large △ !!

SODA 2008

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Note that to prove Griggs-Yeh Conjecture

it suffices to consider regular graphs.

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However….

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Very little was known about exact L(2,1)-labeling numbers for

specific classes of graphs.

--- even for 3-regular graphs

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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

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Other important subclasses of 3-regular graphs

Generalized Petersen Graph

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Generalized Petersen Graph of order 5

GPG(5)

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GPG(3) , GPG(4)

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GPG(6)

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GPG(9)

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Griggs-Yeh Conjecture says that

λ(G) ≦9 for all GPGs G

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Georges and Mauro, 2002, Discrete Math.

λ(G) ≦8 for all GPGs G

except for the Petersen graph

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Georges and Mauro, 2002, Discrete Math.

λ(G) ≦7 for all GPGs G

of order n 6≦ except for

the Petersen graph

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Georges-Mauro Conjecture2002

For any GPG G of order n 7,≧

λ(G) ≦7

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Sarah Spence Adams

Jonathan Cass

Denise Sakai Troxell

2006, IEEE Trans. Circuits & Systems

Georges-Mauro Conjectureis true

for orders 7 and 8

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Generalized Petersen6

grap of order h 6

More….

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Number of non-isomorphic GPGs of order n

with the aid of a computer program

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Georges-Mauro Conjectureis true

for orders 9,10,11 and 12

Y-Z Huang, C-Y Chiang,L-H Huang, H-G Yeh

2009

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Theorem

Generalized Petersen7

gra of order 9ph

Generalized Petersen graphs of orders 9, 10, 11 and 12

One-page proof !!

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3

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3

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1, 2, 4, 5, 6

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3

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Case 1 3

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Case 2 3

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Case 3

3

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Case 4 3

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Case 5 3

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Case 6

3

33

Case 7

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3

33

5 1

6

4

25

1

4

2

1, 2, 4, 5, 60

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Case 1

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3

33

5 1

6

4

25

1

4

2

1, 2, 4, 5, 60

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Case 1Case A

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3

33

0

7

07

0

75 1

6

4

25

1

4

2

1, 2, 4, 5, 60

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Case 1Case A

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3

33

5 1

6

4

25

1

4

2

1, 2, 4, 5, 60

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Case 1Case B

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3

33

7

0

70

7

05 1

6

4

25

1

4

2

1, 2, 4, 5, 60

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Case 1Case B

51

3

33

5 1

6

25

1

4

2

1, 2, 4, 5, 60

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Case 2

4

52

3

33

7

0

70

7

05 1

6

25

1

4

2

1, 2, 4, 5, 60

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Case 2

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Case A

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3

33

0

7

07

0

75 1

6

25

1

4

2

1, 2, 4, 5, 60

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Case 2

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Case B

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3

33

0

7

07

0

76

6

46

2

2

4

2

1, 2, 4, 5, 60

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Case 7

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Theorem

Generalized Petersen7

grap of order h 10

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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

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1

1

1

4, 4, 6

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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

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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 ?

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