On the complexity of a restricted list-coloring problem

We investigate a restricted list-coloring problem. Given a graph G = (V,E), a non empty set of colors L, and a nonempty subset L(v) of L for each vertex v, find an L-coloring of G with the size of each class of colors being equal to a given integer. This restricted list-coloring problem was proposed by de Werra. We prove that this problem is script N sign P-Complete even if the graph is a path with at most two colors on each vertex list. We then give a polynomial algorithm which solves this problem for the case where the total number of colors occurring in all lists is fixed.