  We investigate when a Borel graph admits a (Borel or measurable) orientation
with outdegree bounded by $k$ for various cardinals $k$. We show that for a
p.m.p. graph $G$, a measurable orientation can be found when $k$ is larger than
the normalized cost of the restriction of $G$ to any positive measure subset.
Using an idea of Conley and Tamuz, we can also find Borel orientations of
graphs with subexponential growth; however, for every $k$ we also find graphs
which admit measurable orientations with outdegree bounded by $k$ but no such
Borel orientations. Finally, for special values of $k$ we bound the projective
complexity of Borel $k$-orientability for graphs and graphings of equivalence
relations. It follows from these bounds that the set of equivalence relations
admitting a Borel selector is $\mathbf{\Sigma}^1_2$ in the codes, in stark
contrast to the case of smooth relations.
