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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
