  For a commutative semigroup $S$ with 0, the zero-divisor graph of $S$ denoted
by $\Gamma(S)$ is the graph whose vertices are nonzero zero-divisor of $S$, and
two vertices $x$, $y$ are adjacent in case $xy=0$ in $S$. In this paper we
study the case where the graph $\Gamma(S)$ is complete $r$-partite for a
positive integer $r$. Also we study the commutative semigroups which are
finitely colorable.
