Ping An

For the ring of integers modulo $N$, let $Z_N$ be its set of zero-divisors. The nonzero zero-divisor graph, denoted by $\Gamma(Z_N^*)$, is a simple graph with vertices $Z_N^*=Z_N-\{0\}$ and for distinct $x,y\in Z_N^*$, the vertices $x$ and $y$ are adjacent if and only if $xy=0$. Beck introduced the concept of a zero-divisor graph of a commutative ring. He was mainly interested in characterizing and discussing the rings which are finitely colorable and proved $\chi(R)=clique(R)$ for reduced or principal ideal rings $R$.
An L(2,1)-labelling of a graph $G$ is an integer assignment $f$ to the vertices of $G$ such that:
(i) $|f(x)-f(y)|\geq 2$ if $x$ and $y$ are adjacent, and
(ii)$|f(x)-f(y)|\geq 1$ if the distance of $x$ and $y$ is 2.
The $\lambda$-number $\lambda(G)$ of $G$, is the minimum range over all L(2,1)-labellings. Griggs and Yeh first introduced the problem of L(2,1)-labelling of a graph. They have obtained an upper bound on the $\lambda$-number of a graph $G$ with the maximum degree $\Delta\colon\lambda(G)\leq \Delta^2+2\Delta$ and proposed a conjecture which set the upper bound for $\lambda(G)$ at $\triangle^2$. From then on, many authors have paid attention to the exact values or upper bound of the number $\lambda(G)$. Chang and Kuo proved that $\lambda(G)\leq \triangle^2+\triangle$ and then Kr\'{a}l and \u{S}krekovski improved the upper bound to be $\lambda(G)\leq \triangle^2+\triangle-1$ with maximum degree $\triangle\geq 2$. Grigges and Yeh show that $\lambda(T)$ is either $\triangle+1$, or $\triangle+2$ for a tree $T$ with maximum degree $\triangle\geq 1$. Sakai Troxell characterized the aspects of $(\triangle+2)$-critical trees.
We obtain the exact value of $\lambda$-number of $\Gamma(Z_N^*),\,\lambda(Z_N^*)$, using a new defined equivalence relation ``$\sim$". Note $N=p_1^{n_1}p_2^{n_2}\cdots p_r^{n_r},\ p_10\ {\rm(}i=1,\ldots,r{\rm)},\ p_1,p_2,\ldots,p_r$ are different prime numbers. We show that $\lambda(Z_N^*)=\displaystyle\frac{N}{p_1}+p_1-3$ when $n_1\geq 2$ and $\lambda(Z_N^*)=\displaystyle\frac{N}{p_1}+p_1-2$ when $n_1=1$. Hence we have $\lambda(Z_N^*)=\triangle+p_1-1$ where $\triangle$ is the maximum degree and $p_1$ is the minimum prime number in the prime factorization of $N$. Next we present an L(2,1)-labelling algorithm of $\Gamma(Z_N^*)$.