Perfect Cayley Designs as generalizations of Perfect Mendelsohn Designs

We introduce the concept of a Perfect Cayley Design (PCD) that generalizes that of a Perfect Mendelsohn Design (PMD) as follows. Given an additive group H, a (v, H, 1)-PCD is a pair (X,B) where X is a v-set and B is a set of injective maps from H to X with the property that for any pair (x,y) of distinct elements of X and any h is an element of H-{0} there is exactly one B is an element of B such that B(h')=x, B(h'')=y and h'-h''=h for suitable h',h'' is an element of H. It is clear that a (v,Z(k),1)-PCD simply is a (v, k, 1)-PMD. This generalization has concrete motivations in at least one case. In fact we observe that triplewhist tournaments may be viewed as resolved (v,Z(2)(2),1)-PCD's but not, in general, as resolved (v, 4, 1)-PMD's. We give four composition constructions for regular and 1-rotational resolved PCD's. Two of them make use of difference matrices and contain, as special cases, previous constructions for PMD's by Kageyama and Miao [15] and for Z-cyclic whist tournaments by Anderson, Finizio and Leonard [5]. The other two constructions succeed where sometimes difference matrices fail and their applications allow us to get new PMD's, new Z-cyclic directed whist tournaments and new Z-cyclic triplewhist tournaments. The whist tournaments obtainable with the last two constructions are decomposable into smaller whist tournaments. We show this kind of tournaments useful in practice whenever, at the end of a tournament, some confrontations between ex-aequo players are needed.