The L(2,1)-labelling of
Ping An,
Yinglie Jin,
Nankai University
Z
*
N
An L(2,1)-labelling of a graph G is nonnegative
real-valued function f : V (G)  [0, ) such that :
 (i) |f(x) − f(y)| ≥ 2 if x and y are adjacent,
 (ii)|f(x) − f(y)| ≥ 1 if the distance of x and
y is 2.
The  −number (G) of G, is the minimum
range over all L(2,1)-labellings.
 (G)  min max{ f (v) : v V (G)}
f
Let
K n be a complete graph on n vertices. Then
 ( K n )  2n
Let Pnbe a path on n vertices. Then (i)  ( P2 )  2,
(ii)  ( P3 )   ( P4 )  3, and (iii)  ( Pn )  4 for n  5 .
1
3
0
1
3
0
2
0
3
1
4
0
3
1
Let C n be a cycle of length n. Then  (Cn )  4 for
any n .
Note V (Cn )  {v0 , v1 ,vn1}
(1) If n  0 (mod 3) , then define
0, if i  0 (mod 3)

f (vi )  2, if i  1(mod 3)
4, if i  2 (mod 3)

(2) If n  1(mod 3) ,then redefine the above f at
v n4 ,vn1 as
0,
3,

f (vi )  
1,
4,
if i  n  4
if i  n  3
if i  n  2
if i  n  1
(2) If n  2 (mod 3) ,then redefine the above f at
v n2 and vn 1 as
1, if i  n  2
f (v i )  
3, if i  n  1
Griggs and Yeh proposed a conjecture
 (G)  
2
Griggs and Yeh (1992) obtained an upper bound
 (G)    2
2
Chang and Kuo (1996) proved that
 (G)    
2

Kr a l and Skrekovski (2003) improved the upper bound
to be
'
 (G)      1
2
with maximum degree △ ≥ 2.
The graph ( Z )
*
N
For the ring of integers modulo
*
N , let Z N be its set of nonzero
*
zero-divisors. ( Z N ) is a simple
*
graph with vertices Z N and for
*
distinct x, y  Z N ,the vertices x
and y are adjacent if and only
if xy  0 .
For example : N=15, Z  {3,5,6,9,10,12}
*
N
12
3
5
10
6
( Z )
*
N
9
Note
N  p1n1 p2n2  prnr
p1  p2    pr , ni  0(i  1,, r )
*
Let m1 , m2 be elements of Z N , we define
m1 ~ m2 if (m1 , N )  (m2 , N )
For example:
[ p1 ]  {m : (m, N )  p1 , m  Z N* }
l1 l2
lr
[
p
p

p
Every equivalence class has the form 1 2
r ],
where 0  li  ni , i  1,2,, r and neither li  ni nor
li  0 (i  1,2,r ) can be satisfied simultaneously.
l
l
l
For any equivalence class [ p11 p22  prr ], it is a
clique if
 ni 
li   
2
for i  1,2, r . Otherwise it is
an independent set.
For any equivalence class [n]  [ p1l1 p2l2  prlr ] ,
N
| [n] |    ,
n
where
 is the Euler -funtion.
For example : N=15, Z  {3,5,6,9,10,12}
*
N
[3]={3,6,9,12};3[5]={5,10} 12
5
[3]
10
[5]
6
( Z )
*
N
9
In this paper, we showed that
 ( Z )    p1  1
*
N
Where △ is the maximum degree and p1
is the minimum prime number in the prime
factorization.
N  36  2  3
2
Z
*
N
2
has equivalence classes:
[2]  {2,10,14,22,26,34};
[2 2 ]  {4,8,16,20,28,32};
[3]  {3,15,21,33};
[3 ]  {9,27};
2
[2  3]  {6,30};
[2  32 ]  {18};
[2 2  3]  {12,24}.
 ( Z )  16  2  1  17
*
N
{17}
[2  32 ]1
[2  3]2
[22  3]2
{1,3,5,7,14,15} {8,9,10,11,12,13} {0,2}
{4,6}
[2]6
[ 2 2 ]6
2
[3]4
[3 ]2
{8,9,10,11}
{1,15}
( Z N* )