Ordering of simple connected graph s.t. ordered terms are adjacent to an odd number of vertices

Question

Let $G$ be a simple connected graph. Show that there is an ordering $v_{1}, v_{2}, \ldots , v_{n}$ of $V$ such that at least $\frac{1}{2}(n - 1)$ vertices $v_{j}$ are adjacent to an odd number of vertices $v_{i}$ with $i < j$.