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

Channel-Assignment Problem

Hale, 1980

Hale, 1980, IEEE

1 1

1 1

2 1

1 2 2 2 3 1 3 1 1 3

1 Chromatic number = 3 2 2 2 3 1 3 1 1 3

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.

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

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

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

1 Roberts, 1980 2 2 2 3 1 3 1 1 3

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

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

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

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!

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

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

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

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

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

However….

Very little was known about exact L(2,1)-labeling numbers for specific classes of graphs. --- even for 3-regular graphs

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

Other important subclasses of 3-regular graphs Generalized Petersen Graph

Generalized Petersen Graph of order 5 GPG(5)

GPG(3) , GPG(4)

GPG(6)

GPG(9)

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

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

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

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

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

More….

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

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

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

3 3 3 43 43

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

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

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

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

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

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

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

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

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

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

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

Theorem 55

56

1 Case 1 1 Case 2 1 Case 3 1 Case 4 1 Case 5 1 Case 6 1 Case 7 Case 8 57

1 4, 4, 6 1 1 58

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

Case 8 再次一个 从这开始 次一个 60

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

太过暴力, 不宜在此陈述! . 其余的证明呢?

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