  We give a criterion for a nef divisor $D$ to be semiample on a Calabi--Yau
threefold $X$ when $D^3=0=c_2(X)\cdot D$ and $c_3(X)\neq 0$. As a direct
consequence, we show that on such a variety $X$, if $D$ is strictly nef and
$\nu(D)\neq 1$, then $D$ is ample; we also show that if there exists a nef
non-ample divisor $D$ with $D\not\equiv 0$, then $X$ contains a rational curve
when its topological Euler characteristic is not $0$.
