| Article ID | Journal | Published Year | Pages | File Type |
|---|---|---|---|---|
| 4649896 | Discrete Mathematics | 2009 | 4 Pages |
The graph consisting of the six triples (or triangles) {a,b,c}{a,b,c}, {c,d,e}{c,d,e}, {e,f,a}{e,f,a}, {x,a,y}{x,a,y}, {x,c,z}{x,c,z}, {x,e,w}{x,e,w}, where a,b,c,d,e,f,x,y,za,b,c,d,e,f,x,y,z and ww are distinct, is called a dexagon triple. In this case the six edges {a,c}{a,c}, {c,e}{c,e}, {e,a}{e,a}, {x,a}{x,a}, {x,c}{x,c}, and {x,e}{x,e} form a copy of K4K4 and are called the inside edges of the dexagon triple. A dexagon triple system of order vv is a pair (X,D)(X,D), where DD is a collection of edge disjoint dexagon triples which partitions the edge set of 3Kv3Kv. A dexagon triple system is said to be perfect if the inside copies of K4K4 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 vv with a sub-dexagon triple system of order uu are also sufficient.
