A characterization on orientations of graphs avoiding given lists on out-degrees

Let $G$ be a graph and $F:V(G)\to 2^N$ be a set function. The graph $G$ is said to be \emph{F-avoiding} if there exists an orientation $O$ of $G$ such that $d_O^{+}(v)\mathrm{\notin }F(v)$ for every $v\in V(G)$, where $d_O^{+}(v)$ denotes the out-degree of $v$ in the directed graph $G$ with respect to $O$. In this paper, we give a Tutte-type good characterization to decide the $F$-avoiding problem when for every $v\in V(G)$, $\mathrm{\mid }F(v)\mathrm{\mid }\le \frac{1}{2}(d_G(v)+1)$ and $F(v)$ contains no two consecutive integers. Our proof also gives a simple polynomial algorithm to find a desired orientation. As a corollary, we prove the following result: if for every $v\in V(G)$, $\mathrm{\mid }F(v)\mathrm{\mid }\le \frac{1}{2}(d_G(v)+1)$ and $F(v)$ contains no two consecutive integers, then $G$ is $F$-avoiding. This partly answers a problem proposed by Akbari et. al.(2020)