  An NP-complete coloring or homomorphism problem may become polynomial time
solvable when restricted to graphs with degrees bounded by a small number, but
remain NP-complete if the bound is higher. For instance, 3-colorability of
graphs with degrees bounded by 3 can be decided by Brooks' theorem, while for
graphs with degrees bounded by 4, the 3-colorability problem is NP-complete. We
investigate an analogous phenomenon for digraphs, focusing on the three
smallest digraphs H with NP-complete H-colorability problems. It turns out that
in all three cases the H-coloring problem is polynomial time solvable for
digraphs with degree bounds $\Delta^{+} \leq 1$, $\Delta^{-} \leq 2$ (or
$\Delta^{+} \leq 2$, $\Delta^{-} \leq 1$). On the other hand with degree bounds
$\Delta^{+} \leq 2$, $\Delta^{-} \leq 2$, all three problems are again
NP-complete. A conjecture proposed for graphs H by Feder, Hell and Huang states
that any variant of the $H$-coloring problem which is NP-complete without
degree constraints is also NP-complete with degree constraints, provided the
degree bounds are high enough. Our study is the first confirmation that the
conjecture may also apply to digraphs.
