Perfect T(G) triple systems when G is a matching

Abstract

A T(G) triple is formed by taking a graph G and replacing every edge with a 3-cycle, where all of the new vertices are distinct from all others in G. An edge-disjoint decomposition of 3Kn into T(G) triples is called a T(G) triple system of order n. If we can decompose Kn into copies of a graph G, such that we can form a T(G) triple from each graph in the decomposition and produce a partition of the edges of 3Kn, then the resulting T(G) triple system is called perfect. -- We give necessary and sufficient conditions for the existence of perfect T(G) triple systems when G is a matching with λ edges, which we denote by ⋃λP₂. We then give cyclic perfect decompositions of 3Kn into T(⋃λP₂) triples for all n ≡ 1 (mod 2λ) when λ is even (except for n = 4λ + 1 when λ > 8) as well as completely solve the case λ = 3.