the l(2,1)-labelling of ping an, yinglie jin, nankai university

The L(2,1)-labelling of The L(2,1)-labelling of

Ping An, Yinglie Jin,

Nankai University

*NZ

(i) |ff((xx)) − f − f((yy))|| ≥ 22 if xx and y y are adjacent, (ii)|ff((xx)) − f − f((yy))|| ≥ 11 if the distance of xx and

yy is 2.2.

The −number ((GG)) of GG, is the minimumrange over all L L(2(2,1)1)-labellings.

An An LL(2,1)-labelling of a graph (2,1)-labelling of a graph GG is nonnegative is nonnegative real-valued function such that :real-valued function such that :

)}(:)(max{min)( GVvvfGf

),0[)(: GVf

Let be a complete graph on n vertices. Then nK

nKn 2)(

Let be a path on n vertices. Then (i) ,

(ii) , and (iii) for .

nP 2)( 2 P

3)()( 43 PP 4)( nP 5n

0 3 01 4 3 1

1 3 0 2

1 3 0

Let be a cycle of length n. Then for

any n .

Note

(1) If , then define

nC 4)( nC

},,{)( 110 nn vvvCV

)3(mod0n

)3(mod2,4

)3(mod1,2

)3(mod0,0

)(

iif

iif

iif

vf i

(2) If ,then redefine the above f at

as

)3(mod1n

14 , nn vv

1,4

2,1

3,3

4,0

)(

niif

niif

niif

niif

vf i

(2) If ,then redefine the above f at

and as

)3(mod2n

2nv 1nv

1,3

2,1)(

niif

niifvf i

Griggs and Yeh proposed a conjecture

2)( G

Griggs and Yeh (1992) obtained an upper bound

Chang and Kuo (1996) proved that

(2003) improved the upper bound to be

with maximum degree △ ≥ 2.

2)( 2G

2)(G

1)( 2 G

krekovskiS and laKr'

The graph

For the ring of integers modulo N , let be its set of nonzero zero-divisors. is a simple graph with vertices and for distinct ,the vertices x and y are adjacent if and only if .

)( *NZ

)( *NZ

*NZ

*NZ

*, NZyx

0xy

}12,10,9,6,5,3{* NZFor example : N=15,

3

6

5

9

10

12

)( *NZ

Let , be elements of , we define

2m1m *NZ

Note

),,1(0,21

2121

rinppp

pppN

ir

nr

nn r

),(),( if ~ 2121 NmNmmm

},),(:{][ *11 NZmpNmmp

For example:

Every equivalence class has the form ,where and neither nor can be satisfied simultaneously.

][ 2121

rlr

ll ppp rinl ii ,,2,1,0 ii nl

),2,1(0 rili

For any equivalence class , it is a

clique if for . Otherwise it is

an independent set.

][ 2121

rlr

ll ppp

ri ,2,1

2i

i

nl

For any equivalence class ,

, where is the Euler -funtion.

][][ 2121

rlr

ll pppn

n

Nn |][|

}12,10,9,6,5,3{* NZFor example : N=15,

3

6

5

9

10

12

)( *NZ

[3]={3,6,9,12}; [5]={5,10}

[5]

[3]

1)( 1* pZN

In this paper, we showed that

Where is the maximum degree and △is the minimum prime number in the prime factorization.

1p

has equivalence classes:

22 3236 N*NZ

}.24,12{]32[

};18{]32[

};30,6{]32[

};27,9{]3[

};33,21,15,3{]3[

};32,28,20,16,8,4{]2[

};34,26,22,14,10,2{]2[

2

2

2

2

)( *NZ

12 ]32[

6]2[ 62 ]2[ 2]32[ 2

2 ]32[

4]3[ 22 ]3[

{8,9,10,11,12,13}

{1,3,5,7,14,15}

{8,9,10,11} {1,15}

{17}

{0,2} {4,6}

171216)( * NZ