| Abstract: | Let v, k, and n be positive integers. An incomplete perfect Mendelsohn design, denoted by k-IPMD(v, n), is a triple (X, Y, ??) where X is a v-set (of points), Y is an n-subset of X, and ?? is a collection of cyclically ordered k-subsets of X (called blocks) such that every ordered pair (a, b) ∈ (X × X)\(Y × Y) appears t-apart in exactly one block of ?? and no ordered pair (a,b) ∈ Y × Y appears in any block of ?? for any t, where 1 ≤ t ≤ k ? 1. In this article, the necessary conditions for the existence of a 4-IPMD(v, n), namely (v ? n) (v ? 3n ? 1) ≡ 0 (mod 4) and v ≥ 3n + 1, are shown to be sufficient for the case n = 3. For the case n = 2, these conditions are sufficient except for v = 7 and with the possible exception of v = 14,15,18,19,22,23,26,27,30. The latter result provides a useful application to the problem relating to the packing of perfect Mendelsohn designs with block size 4. © 1994 John Wiley & Sons, Inc. |
