Doyen-Wilson theorem for perfect hexagon triple systems

作者：Gao, Hua; Wang, Jinhua^*

来源：Discrete Mathematics, 2011, 311(12): 1006-1014.

DOI：10.1016/j.disc.2011.03.004

• Hexagon triple system
• Group divisible hexagon triple system
• Embedding
• Subsystem

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! 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), d, e), and (e, f, a) are called outside 3-cycles. A hexagon triple system of order v is a pair (X, l), where l is a collection of edge disjoint hexagon triples which partitions the edge set of 3K(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, Kucukcifci and Lindner had shown that there is a perfect hexagon triple system of order v if and only if v equivalent to 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 >= 2u + 1, v, u equivalent to 1, 3(mod 6) and u >= 7, which is a perfect hexagon triple system analogue of the Doyen Wilson theorem.

• 出版日期2011-6-28
• 单位南通大学