The L(2,1)-labelling of Ping An, Yinglie Jin, Nankai University

f(x) − f(y)|2xy (i) | f(x) − f(y)| ≥ 2 if x and y are adjacent, f(x) − f(y)|1x y 2. (ii)| f(x) − f(y)| ≥ 1 if the distance of x and y is 2. (G)G The −number (G) of G, is the minimum L(21) range over all L(2, 1) -labellings. An L(2,1)-labelling of a graph G is nonnegative real-valued function such that :

Let be a complete graph on n vertices. Then Let be a path on n vertices. Then (i), (ii), and (iii) for

Let be a cycle of length n. Then for any n. Note (1) If, then define (2) If,then redefine the above f at as

(2) If,then redefine the above f at and as

Griggs and Yeh proposed a conjecture

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.

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.

For example : N=15,

Let, be elements of, we define Note For example:

Every equivalence class has the form, where and neither nor can be satisfied simultaneously. For any equivalence class, it is a clique if for. Otherwise it is an independent set. For any equivalence class,, where is the Euler -funtion.

For example : N=15, [3]={3,6,9,12}; [5]={5,10} [5] [3]

In this paper, we showed that Where △ is the maximum degree and is the minimum prime number in the prime factorization.

has equivalence classes:

{8,9,10,11,12,13}{1,3,5,7,14,15} {8,9,10,11} {1,15} {17} {0,2} {4,6}