Existence for Perfect \(T(K_{1,k})\)-triple Systems

Abstract

Let \(G\) be a subgraph of \(K_n\). 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 \(3K_n\) 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 \(K_n\), 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, Kıvcıkgızı, and Rosa determined the spectra for cases where \(G\) is any subgraph of \(K_4\). In this paper, we will focus on the star graph \(K_{1,k}\) and discuss the existence of perfect \(T(K_{1,k})\)-triple systems. Especially, for prime powers \(k\), its spectra are completely determined.