Prove that if the vertex set of a directed graph can be partitioned into three subsets \(V_{1}, V_{2}\), and \(V_{3}\) such that edges only exist from \(V_{1}\) into \(V_{2}\), or from \(V_{2}\) into \(V_{3}\), or from \(V_{3}\) into \(V_{1}\), then the graph is not regular. Give an example of such a graph with eight vertices.