Perfect dexagon triple systems with given subsystems

The graph consisting of the six triples (or triangles) {a, b, c}, {c, d, e}, {e, f, a}, {x, a, y}, {x, c, z}, {x, e, w}, where a, b, c. cl, e,f, x, y, z and w are distinct, is called a dexagon triple. In this case the six edges {a, c}, {c, e}, {e, a}, {x, a}, {x, c}, and {x, e} form a copy of K(4) and are called the inside edges of the dexagon triple. A dexagon triple system of order v is a pair (X, D), where D is a collection of edge disjoint dexagon triples which partitions the edge set of 3K(v). A dexagon triple system is said to be perfect if the inside copies of K(4) form a block design. In this note, we investigate the existence of a dexagon triple system with a Subsystem. We show that the necessary conditions for the existence of a dexagon triple system of order v with a sub-dexagon triple system of order u are also sufficient. (C) 2008 Elsevier B.V. All rights reserved.