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|<12d, 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.