Minimum Degree and the Orientation of a Graph

Abstract

Let $G$ be a simple graph such that $\delta (G)\ge \lfloor \frac{|V(G)|}{2}\rfloor +k$, where $k$ is a non-negative integer, and let $f:V(G)\to {\mathbb{Z}}^{+}$ be a function having the following properties (i)$\frac{d_G(x)}{2}-\frac{k+1}{2}\le f(x)\le \frac{d_G(x)}{2}+\frac{k+1}{2}$ for every $x\in V(G)$, (ii)$\sum _{x\in V(G)}f(x)=|E(G)|$. Then $G$ has an orientation $D$ such that $d_D^{+}(x)=f(x)$, for every $x\in V(G)$.