Abstarct: Given an edge coloring with colors {1, 2, , k}, we call a vertex coloring of G with colors {1, 2, , k} adaptable to the edge coloring if for any edge of G, the colors on the end vertices of it and the edge itself are different. The smallest k for which there exists a adaptable coloring for any edge coloring with colors {1, 2, , k} is called the adaptable coloring number of G, denoted by χad(G) . In this thesis, we describe some new results in the concept of the adaptable coloring of graphs. Also We define the adaptable choice number graphs and we study some results concerning the choice number of graphs.