Uniformly resolvable decompositions of λ-fold complete multipartite graph into 4-star

Let ?Ku [1] be the ?-fold complete multipartite graph with u parts of size 1. A (K1,n, ?)-resolvable group divisible design (RGDD) of type gu is a K1,n-decomposition of the graph ?Ku [g] into parallel classes each of which is a partition of the vertex set. A (K1,n, ?)-frame of type gu is a K1,n-decomposition of ?Ku [g] into partial parallel classes each of which is a partition of the vertex set except for those vertices in one of the u parts. In this paper, we completely solve the existence of a (K1,4, ?)-frame and a (K1,4, ?)-RGDD of type gu for any admissible parameters g, u and ?.