The coloration of graphs having vertices preassigned to specific colors is discussed. Also considered is the case of graphs which have vertices that are not to be assigned to specified colors, with the colors being dependent upon each particular vertex. The theorems proved show that a graph with these constraints reduces to a graph without such constraints. Furthermore, each constrained graph is colorable with a given number of colors if and only if the unconstrained graph is colorable with the same number of colors. Using the results, a formal relationship is established between the problem of determining the existence and the problem of determining an n-coloration of a graph.