The Asymptotic Existence of Resolvable Group Divisible Designs

Authors


Corresponding Author Peter Dukes, E-mail: dukes@uvic.ca; Research of Peter Dukes is supported by NSERC.

Abstract

A group divisible design (GDD) is a triple math formula which satisfies the following properties: (1) math formula is a partition of X into subsets called groups; (2) math formula is a collection of subsets of X, called blocks, such that a group and a block contain at most one element in common; and (3) every pair of elements from distinct groups occurs in a constant number λ blocks. This parameter λ is usually called the index. A k-GDD of type math formula is a GDD with block size k, index math formula, and u groups of size g. A GDD is resolvable if the blocks can be partitioned into classes such that each point occurs in precisely one block of each class. We denote such a design as an RGDD. For fixed integers math formula and math formula, we show that the necessary conditions for the existence of a k-RGDD of type math formula are sufficient for all math formula. As a corollary of this result and the existence of large resolvable graph decompositions, we establish the asymptotic existence of resolvable graph GDDs, G-RGDDs, whenever the necessary conditions for the existence of math formula-RGDs are met. We also show that, with a few easy modifications, the techniques extend to general index. © 2012 Wiley Periodicals, Inc. J. Combin. Designs 21: 112–126, 2013

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