Orientations of graphs avoiding given lists on out‐degrees

Let be a graph and be a mapping. The graph is said to be ‐ avoiding if there exists an orientation of such that for every , where denotes the out‐degree of in the directed graph with respect to . In this paper it is shown that if is bipartite and for every , then is ‐avoiding. The bound is best possible. For every graph , we conjecture that if for every , then is ‐avoiding. We also argue about this conjecture for the best possibility of the conditions and also show some partial solutions.