Let  $R$  be a Noetherian ring and let  $Z(R)$  be the set of all  zero-divisors of $R$.  
We denote by  $G(R)$  the simple graph whose vertices are elements of  $R$ 
and in which two distinct vertices  $x$  and  $y$  are joined by an edge if  $x - y$  is in  $Z(R)$. 
Let  $\chi(R)$  be the chromatic number of the graph  $G(R)$.  
If  $\chi(R)$  is finite, then  $R$  is an integral domain or  $R$  is a finite Artin ring.  
In the former case we have $\chi(R) = 1$  and in the latter case we get  $\chi(R) = {\rm max}\{|M_{1}|,
\ldots,|M_{t}|\}$  where  $M_{1}, \ldots, M_{t}$  are all maximal ideals of 
$R$  and  $|M_{i}|$  denotes the number of elements of the set  $M_{i}$  for $i = 1, \ldots, t$.