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

Abstract

Let λKu[g] be the λ-fold complete multipartite graph with u parts of size g. A (K1,n, λ)-resolvable group divisible design (RGDD) of type g u is a K1,n-decomposition of the graph λKu[g] into parallel classes each of which is a partition of the vertex set. A (K1,n, λ)-frame of type g u is a K1,n-decomposition of λKu[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, λ)-frame and a (K1,4, λ)-RGDD of type g u for any admissible parameters g, u and λ.