For a nontrivial connected graph G, center coloring is a kind of coloring that is to color the vertices of a graph G is such a way that if vertices have different distance from the center then they must receive different colors. Two adjacent vertices can receive the same color. The number of colors required of such a coloring is called center coloring number Cc ( G) of G. [ 7] This coloring can be applied to hierarchy problems to find the number of structures, people, criteria and comparisons, etc. Moreover it can be applied to earthquake motion problems to find the number of settlements that are affected by an earthquake. The center coloring number of some well-known classes of graphs are determined and several bounds are established for the center coloring number of a graph in terms of other graphical parameters.