Perfect \(T(G)\)-Triple System for Each Graph G with Five Vertices and Seven Edges

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, Kıygıkçifi, and Rosa determined the spectra for cases where \(G\) is any subgraph of \(K_4\). Then, in our previous paper, the spectrum of perfect \(T(G)\)-triple systems for each graph \(G\) with five vertices and \(i (\leq 6)\) edges was determined. In this paper, we will completely solve the spectrum problem of perfect \(T(G)\)-triple systems for each graph \(G\) with five vertices and seven edges.