Are there any results on coloring a graph using a limited number of each color. In other words, the decision problem would be: given a list of colors $C = (c_1, \dots, c_k)$ where each color $c_i$ is associated with a bound on the number of times it can be used $b_i$ (where there can be at most $b_i$ nodes colored with color $c_i$) can you color the vertices of a graph where no two adjacent vertices are colored the same color. This decision problem is NP-hard when there are no constraints on the number of times a color is used so it is also hard in this setting.
But has this problem been studied in graphs where chromatic number is easy such as interval graphs or perfect graphs? It is not clear to me that this problem is easy on graph classes where finding the chromatic number is polynomial time.