Vertex coloring is the following optimization problem; given a graph, how many colors are required to color its vertices in such a way that no two adjacent vertices receive the same color? The required number of colors is called the chromatic number of G and is denoted by (G). In this thesis, we reviewed the vertex coloring concepts and theorems. The package ColorG which we have improved has many functions for dealing with graph coloring. This package uses a heuristic method due to Brelaz to color the graph so that adjacent vertices have distinct colors.