Doyen–Wilson theorem for perfect hexagon triple systems

The graph consisting of the three 3 -cycles (or triples) ( a , b , c ) , ( c , d , e ) , and ( e , f , a ) , where a , b , c , d , e and f are distinct is called a hexagon triple. The 3 -cycle ( a , c , e ) is called an inside 3 -cycle; and the 3 -cycles ( a , b , c ) , ( c , d , e ) , and ( e , f , a ) are called outside 3 -cycles. A hexagon triple system of order v is a pair ( X , C ) , where C is a collection of edge disjoint hexagon triples which partitions the edge set of 3 K v . Note that the outside 3 -cycles form a 3 -fold triple system. If the hexagon triple system has the additional property that the collection of inside 3 -cycles ( 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 v if and only if v ≡ 1 , 3 ( mod 6 ) and v ≥ 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 v with a perfect sub-hexagon triple system of order u if and only if v ≥ 2 u + 1 , v , u ≡ 1 , 3 ( mod 6 ) and u ≥ 7 , which is a perfect hexagon triple system analogue of the Doyen–Wilson theorem.