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 , d , 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 3 K 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.