This analysis determines the existence of (K1,4, ?) frames and RGDDs in complete multipartite graphs, showing the conditions required for their formation.
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 ?.
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Wang Li (2025) studied this question.
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