Orientations of graphs avoiding given lists on out‐degrees

Let G be a graph and F:V(G)-> 2N be a mapping. The graph G is said to be F-avoiding if there exists an orientation O of G such that dO+(v)is not an element of F(v) for every v is an element of V(G), where dO+(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 divide F(v) divide <= dG(v)/2 for every v is an element of V(G), then G is F-avoiding. The bound divide F(v) divide <= dG(v)/2 is best possible. For every graph G, we conjecture that if divide F(v) divide <= 12(dG(v)-1) for every v is an element of 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.