Doyen–Wilson theorem for perfect hexagon triple systems

The graph consisting of the three 33-cycles (or triples) (a,b,c)(a,b,c), (c,d,e)(c,d,e), and (e,f,a)(e,f,a), where a,b,c,d,ea,b,c,d,e and ff are distinct is called a hexagon triple. The 33-cycle (a,c,e)(a,c,e) is called an inside 33-cycle; and the 33-cycles (a,b,c)(a,b,c), (c,d,e)(c,d,e), and (e,f,a)(e,f,a) are called outside 33-cycles. A hexagon triple system of order vv is a pair (X,C)(X,C), where CC is a collection of edge disjoint hexagon triples which partitions the edge set of 3Kv3Kv. Note that the outside 33-cycles form a 33-fold triple system. If the hexagon triple system has the additional property that the collection of inside 33-cycles (a,c,e)(a,c,e) is a Steiner triple system it is said to be perfect. In 2004, Küçükçifçi and Lindner had shown that there is a perfect hexagon triple system of order vv if and only if v≡1,3(mod6) and v≥7v≥7. In this paper, we investigate the existence of a perfect hexagon triple system with a given subsystem. We show that there exists a perfect hexagon triple system of order vv with a perfect sub-hexagon triple system of order uu if and only if v≥2u+1v≥2u+1, v,u≡1,3(mod6) and u≥7u≥7, which is a perfect hexagon triple system analogue of the Doyen–Wilson theorem.