Partial Steiner triple systems with equal-sized holes
type: Item
instance of: scholarly article
title: Partial Steiner triple systems with equal-sized holes
zbMATH Open document ID: 0818.05013
DOI: 10.1016/0097-3165(95)90080-2
author: Charles A. Cusack
author: Charles J. Colbourn
author: Donald L. Kreher
published in: Journal of Combinatorial Theory. Series A
publication date: 1995-04-04T00:00:00Z
review text: In this elegant paper the authors completely dispose of the existence problem for 3-GDDs of type \(u^ r l^ t\), i.e. of group divisible designs with \(r\) groups of size \(u\) and \(t\) groups of size \(l\), and all blocks of size 3 (here \(u\), \(r\), \(t\) are positive integers). According to their main theorem, such a GDD exists if and only if the following conditions are satisfied: (i) \(u\) is odd, (ii) \(r+ t\) is odd, (iii) if \(r= 1\), \(t\geq u+1\), (iv) if \(r= 2\), then \(t\geq u\), and (v) \(\left(\begin{smallmatrix} t\\ 2\end{smallmatrix}\right)+ rut+ \left(\begin{smallmatrix} r\\ 2\end{smallmatrix}\right) u^ 2\equiv 0\pmod 3\). Their starting point is a theorem in [the first author, \textit{D. G. Hoffman} and \textit{R. Rees}, A new class of group divisible designs with block size three, J. Comb. Theory, Ser. A 59, No. 1, 73-89 (1992; Zbl 0759.05012)] as well as the computational results in [the first author, Small group divisible designs with block size three, J. Comb. Math. Comb. Comput. 14, 153-172 (1993; Zbl 0793.05012)]. Their methods involve the full arsenal of design- theoretical tools, including resolvable GDDs and their variations, transversal designs, and variants of Wilson's fundamental construction.
Mathematics Subject Classification ID: 05B07
Mathematics Subject Classification ID: 05B05
zbMATH DE Number: 740287
zbMATH Keywords: partial Steiner triple systems
zbMATH Keywords: existence
zbMATH Keywords: group divisible designs
reviewed by: Alexander Rosa
MaRDI profile type: Publication
full work available at URL: 0097-3165(95)90080-2
OpenAlex ID: W2093576151
cites work: Q4190650
cites work: Q4288081
cites work: A new class of group divisible designs with block size three
cites work: Uniformly resolvable pairwise balanced designs with blocksizes two and three
cites work: Two new direct product‐type constructions for resolvable group‐divisible designs
cites work: Q4093450
cites work: Q4026152
description: scientific article; zbMATH DE number 740287