Outline and amalgamated triple systems of even index (Q2766439)

A triple system is an edge-partition of the complete multigraph into triangles; the number of times each vertex pair appears as an edge is called the index of the triple system. An amalgamation of such a triple system is obtained by repeatedly identifying vertices without identifying or deleting edges, while retaining the initial edge-partition. Elementary conditions on the vertex degrees and edge-partition can be deduced to indicate when a multigraph with loops might be an amalgamated triple system; such a multigraph is called an outline triple system when it satisfies the five conditions so deduced. In this paper, a detailed proof of the statement that, when the index is even, every outline triple system is an amalgamation of a triple system, is given. This gives a fundamental structural characterization of triple systems with even index, and that, for example, gives an elegant structural result on embeddings of partial triple systems.