Existence for perfect T(K1, k)-triple systems

Let G be a subgraph of Kn. The graph obtained from G by replacing each edge with a 3-cycle whose third vertex is distinct from other vertices in the configuration is called a T (G)-triple. An edge-disjoint decomposition of 3Kn into copies of T (G) is called a T (G)-triple system of order n. If, in each copy of T (G) in a T (G)triple system, one edge is taken from each 3-cycle (chosen so that these edges form a copy of G) in such a way that the resulting copies of G form an edge-disjoint decomposition of Kn, then the T (G)triple system is said to be perfect. The set of positive integers n for which a perfect T (G)-triple system exists is called its spectrum. Earlier papers by authors including Billington, Lindner, Küçükçifçi and Rosa determined the spectra for cases where G is any subgraph of K4. In this paper, we will focus in star graph K1,k and discuss the existence for perfect T (K1,k)-triple system. Especially, for prime powers k, its spectra are completely determined.