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

Let lambda K-u[g] be the A-fold complete multipartite graph with u parts of size g. A (K-1,K-n, lambda)-resolvable group divisible design (RGDD) of type g(u) is a K-1,K-n-decomposition of the graph lambda K-u[g] into parallel classes each of which is a partition of the vertex set. A (K-1,K-n, lambda)-frame of type g(u) is a K-1,K-n-decomposition of lambda K-u[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 (K-1,K-4, lambda)-frame and a (K-1,K-4, lambda)-RGDD of type g(u )for any admissible parameters g, u and lambda.