  We prove a descriptive version of Brooks's theorem for directed graphs. In
particular, we show that, if $D$ is a Borel directed graph on a standard Borel
space $X$ such that the maximum degree of each vertex is at most $d \geq 3$,
then unless $D$ contains the complete symmetric directed graph on $d + 1$
vertices, $D$ admits a $\mu$-measurable $d$-dicoloring with respect to any
Borel probability measure $\mu$ on $X$, and $D$ admits a
$\tau$-Baire-measurable $d$-dicoloring with respect to any Polish topology
$\tau$ compatible with the Borel structure on $X$. We also prove a definable
version of Gallai's theorem on list dicolorings for directed graphs by showing
that any Borel directed graph of bounded degree whose connected components are
not Gallai trees is Borel degree-list-dicolorable.
