The existence of perfect T(K1,2p)-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 et al. determined the spectra for cases where G is any subgraph of K4. In this paper, we will focus on star graphs K1,2p and discuss the existence for perfect T(K1,2p)-triple systems. In particular, for odd prime power p, the spectrum is completely determined.