Minimum Degree and the Orientation of a Graph

Abstract

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