Decompositions of complete tripartite graphs into triangles with an edge attached

Abstract

Let K (r, s, t) denote the complete tripartite graph with partite sets of size r, s and t, where r ≤ s ≤ t. Let D be the graph consisting of a triangle with an edge attached. We show that K(r, s, t) may be decomposed into copies of D if and only if 4 divides rs + st + rt and t ≤ 3rs/(r + s).