Perfect dexagon triple systems (Q2468000)

A dexagon triple is a graph that is isomorphic to the union of the six triangles \(\{a,b,e\}\), \(\{a,c,g\}\), \(\{a,d,h\}\), \(\{b,c,f\}\), \(\{b,d,i\}\), and \(\{c,d,j\}\); the six edges induced on \(\{a,b,c,d\}\) by the dexagon triple form an ``interior'' complete graph \(K_4\). A dexagon triple system of order \(n\) is a partition of the edges of the three-fold copy of the complete graph, \(3K_n\), into dexagon triples. A dexagon triple system is perfect if the set of ``interior'' \(K_4\)s of the dexagon triples form a block design with block size four (and index one). It is shown that a perfect dexagon triple system of order \(n\) exists if and only if \(n \equiv 1 \pmod{12}\).