Tuyển tập các báo cáo nghiên cứu khoa học trên tạp chí toán học quốc tế đề tài:Class-Uniformly Resolvable Group Divisible Structures I: Resolvable Group Divisible Designs. | Class-Uniformly Resolvable Group Divisible Structures I Resolvable Group Divisible Designs Peter Danziger Department of Mathematics Physics and Computer Science Ryerson Polytechnic University Toronto ON M5B 2K3 Canada danziger@ Brett Stevens School of Mathematics and Statistics Carleton University 1125 Colonel By Dr. Ottawa ON K1S 5B6 Canada brett@ Submitted Jun 3 2003 Accepted Mar 15 2004 Published Mar 25 2004 MR Subject Classifications 05B05 05B40 Abstract We consider Class-Uniformly Resolvable Group Divisible Designs CURGDD which are resolvable group divisible designs in which each of the resolution classes has the same number of blocks of each size. We derive the fully general necessary conditions including a number of extremal bounds. We present some general constructions including a novel construction for shrinking the index of a master design. We construct a number of infinite families primarily with block sizes 2 and k including some extremal cases. 1 Introduction Class-Uniformly Resolvable incidence structures where each resolution class has the same number of blocks of each size are discussed in 3 4 5 12 . These references contain motivations applications and discussions of related objects. We will assume that the reader is acquainted with design theory terminologies and we refer them to 2 . In this article we investigate class-uniformly resolvable group divisible designs Supported by NSERC discovery grant OGP0170220. Supported by PIMS MITACS and IBM Watson Research and NSERC. THE ELECTRONIC JOURNAL OF COMBINATORICS 11 2004 R23 1 Definition . A Class-Uniformly Resolvable Group Divisible Design CURGDDx of type n gu with partition JJ kPk is a GDDÀ with the additional property that the blocks can be partitioned into resolution classes with partition JJ kpk. CURGDDs were introduced by Lamken et al. and were used by Wevrick and Vanstone to construct CURDs 5 12 . A CURGDD with all g 1 is a CURD. Also a CURD with partition kpk1 .
