Orientations of graphs avoiding given lists on out‐degrees

Let G be a graph and F : V ( G ) → 2 N be a mapping. The graph G is said to be F‐ avoiding if there exists an orientation O of G such that d O + ( v ) ∉ F ( v ) for every v ∈ V ( G ), where d O + ( v ) denotes the out‐degree of v in the directed graph G with respect to O. In this paper it is shown that if G is bipartite and ∣ F ( v ) ∣ ≤ d G ( v ) / 2 for every v ∈ V ( G ), then G is F‐avoiding. The bound ∣ F ( v ) ∣ ≤ d G ( v ) / 2 is best possible. For every graph G, we conjecture that if ∣ F ( v ) ∣ ≤ 1 2 ( d G ( v ) − 1 ) for every v ∈ V ( G ), then G is F‐avoiding. We also argue about this conjecture for the best possibility of the conditions and also show some partial solutions.