The Asymptotic Existence of Resolvable Group Divisible Designs

• Published in: Journal of combinatorial designs 2013-03, Vol.21 (3), p.112-126
• Main Authors: Chan, Justin H., Dukes, Peter J., Lamken, Esther R., Ling, Alan C.H.
• Format: Article
• Language: English
• Subjects: frame design; graph decomposition; group divisible design; resolvable design; Studies
• ISSN: 1063-8539; 1520-6610
• DOI: 10.1002/jcd.21315

A group divisible design (GDD) is a triple (X,G,B) which satisfies the following properties: (1) G is a partition of X into subsets called groups; (2) B 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 gu is a GDD with block size k, index λ=1, 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 g≥1 and k≥2, we show that the necessary conditions for the existence of a k‐RGDD of type gu are sufficient for all u≥u0(g,k). 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 (v,G,1)‐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