A brief survey of perfect Mendelsohn packing and covering designs Original Research Article

A (v,k,λ)-perfect Mendelsohn packing (covering) design is a pair (X,A) where X is a v-set of points and A is a collection of cyclically ordered k-subsets of X, called blocks, such that every ordered pair of points of X appears t-apart in at most (at least) λ blocks of A for all t=1,2,…,k−1. If no other such packing (covering) has more (fewer) blocks, the packing (covering) is said to be maximum (minimum) and the number of blocks in a maximum packing (a minimum covering) is called the packing number (the covering number), denoted by P(v,k,λ) (C(v,k,λ)). The packing problem is to determine the values of P(v,k,λ) for all v⩾k. Similarly, the covering problem is to determine the values of C(v,k,λ) for all v⩾k. The values of the functions P(v,3,λ) and C(v,3,λ) have been determined completely for all v⩾3 and λ⩾1. The functions P(v,4,1) and C(v,4,1) have been determined for all v⩾4, to within a handful of possible exceptions. For k=5 and λ=1, the packing problem has been solved, with a few possible exceptions, but the covering problem remains to be investigated. We shall present a brief survey of the current results and mention some open problems. For the most part, the main results have been established on the strength of the existence of certain types of incomplete perfect Mendelsohn designs (IPMDs), which are of interest in their own right.