Perfect Cayley Designs as Generalizations of Perfect Mendelsohn Designs

AbstractWe introduce the concept of a Perfect Cayley Design(PCD) that generalizes that of a Perfect Mendelsohn Design (PMD) as follows. Given anadditive group H, a (v, H, 1)-PCDis a pair $$(X,\mathcal{B})$$ whereX is a v-set and $$\mathcal{B}$$ isa set of injective maps fromH toX with the property that for any pair (x,y)of distinct elements of X and any h ∈ H - {0} there is exactly one B ∈ $$\mathcal{B}$$ such that B(h')=x, B(h'')=yandh'-h''=h for suitable h',h'' ∈ H.It is clear that a (v,Z_k,1)-PCD simply is a(v, k, 1)-PMD.This generalization has concretemotivations in at least one case. In fact we observe thattriplewhist tournaments may be viewed as resolved(v,Z22,1)-PCD's but not, in general, as resolved(v, 4, 1)-PMD's.We give four composition constructionsfor regular and 1-rotational resolved PCD's. Two of them make use of differencematrices and contain, asspecial cases, previous constructions for PMD's by Kageyama andMiao [15] and for Z-cyclic whist tournaments by Anderson,Finizio and Leonard [5]. The other two constructions succeed wheresometimes difference matrices fail and their applications allow us to get new PMD's, new Z-cyclic directed whist tournaments and newZ-cyclic triplewhist tournaments.The whist tournaments obtainable with the last twoconstructions are decomposable into smaller whist tournaments.We show this kind of tournaments useful in practice whenever, at theend of a tournament, some confrontations between ex-aequo players areneeded.