  For positive integers $j\ge k$, an $L(j,k)$-labeling of a digraph $D$ is a
function $f$ from $V(D)$ into the set of nonnegative integers such that
$|f(x)-f(y)|\ge j$ if $x$ is adjacent to $y$ in $D$ and $|f(x)-f(y)|\ge k$ if
$x$ is of distant two to $y$ in $D$. Elements of the image of $f$ are called
labels. The $L(j,k)$-labeling problem is to determine the
$\vec{\lambda}_{j,k}$-number $\vec{\lambda}_{j,k}(D)$ of a digraph $D$, which
is the minimum of the maximum label used in an $L(j,k)$-labeling of $D$. This
paper studies $\vec{\lambda}_{j,k}$- numbers of digraphs. In particular, we
determine $\vec{\lambda}_{j,k}$- numbers of digraphs whose longest dipath is of
length at most 2, and $\vec{\lambda}_{j,k}$-numbers of ditrees having dipaths
of length 4. We also give bounds for $\vec{\lambda}_{j,k}$-numbers of bipartite
digraphs whose longest dipath is of length 3. Finally, we present a linear-time
algorithm for determining $\vec{\lambda}_{j,1}$-numbers of ditrees whose
longest dipath is of length 3.
