Need efficient correct high level algorithm description with correctness proof.

A verter-coloring of a graph is an assignment of a color to each vertex. Given a directed graph G (with no self-loops) design an algorithm that determines the minimum number of colors necessary to make a verter-coloring of G with the restriction that two vertices that are in the same cycle cannot be the same color. For example, this graph can be colored using 3 different colors. This turns out to be the minimum number of colors for this graph