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 $i(\le 6)$ 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.