On orientations with forbidden out-degrees

Let G be a d-regular graph and let F⊆\0, 1, 2, …, d\ be a list of forbidden out-degrees. Akbari, Dalirrooyfard, Ehsani, Ozeki, and Sherkati conjectured that if |F|<1/2d, then G should admit an F-avoiding orientation, i.e., an orientation where no out-degrees are in the forbidden list F. The conjecture is known for d≤ 4 due to work of Ma and Lu, and here we extend this to d≤ 6. The conjecture has also been studied in a generalized version, where $d, F$ are changed from constant values to functions $d(v), F(v)$ that vary over all v∈ V(G). We provide support for this generalized version by verifying it for some new cases, including when G is 2-degenerate and when every $F(v)$ has some specific structure.