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Incomplete perfect mendelsohn designs with block size 4 and holes of size 2 and 3

✍ Scribed by F. E. Bennett; H. Shen; J. Yin

Publisher
John Wiley and Sons
Year
1994
Tongue
English
Weight
643 KB
Volume
2
Category
Article
ISSN
1063-8539

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✦ Synopsis

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, 𝔹) where X is a v‐set (of points), Y is an n‐subset of X, and 𝔹 is a collection of cyclically ordered k‐subsets of X (called blocks) such that every ordered pair (a, b) ∈ (X Γ— X)(Y Γ— Y) appears t‐apart in exactly one block of 𝔹 and no ordered pair (a,b) ∈ Y Γ— Y appears in any block of 𝔹 for any t, where 1 ≀ t ≀ k βˆ’ 1. In this article, the necessary conditions for the existence of a 4‐IPMD(v, n), namely (v βˆ’ n) (v βˆ’ 3n βˆ’ 1) ≑ 0 (mod 4) and v β‰₯ 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. Β© 1994 John Wiley & Sons, Inc.

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