{"title":"An update on the existence of Kirkman triple systems with steiner triple systems as subdesigns","authors":"P. Dukes, E. Lamken","doi":"10.1002/jcd.21844","DOIUrl":null,"url":null,"abstract":"A Kirkman triple system of order v , KTS (v) , is a resolvable Steiner triple system on v elements. In this paper, we investigate an open problem posed by Doug Stinson, namely the existence of KTS (v) which contain as a subdesign a Steiner triple system of order u , an STS (u) . We present several different constructions for designs of this form. As a consequence, we completely settle the extremal case v= 2u+ 1 , for which a list of possible exceptions had remained for close to 30 years. Our new constructions also provide the first infinite classes for the more general problem. We reduce the other maximal case v= 2u+ 3 to (at present) three possible exceptions. In addition, we obtain results for other cases of the form v= 2u+w and also near v= 3u . Our primary method introduces a new type of Kirkman frame which contains group divisible design subsystems. These subsystems can occur with different configurations, and we use two different varieties in our constructions.","PeriodicalId":15389,"journal":{"name":"Journal of Combinatorial Designs","volume":"30 1","pages":"581 - 608"},"PeriodicalIF":0.5000,"publicationDate":"2022-05-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Combinatorial Designs","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1002/jcd.21844","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
Abstract
A Kirkman triple system of order v , KTS (v) , is a resolvable Steiner triple system on v elements. In this paper, we investigate an open problem posed by Doug Stinson, namely the existence of KTS (v) which contain as a subdesign a Steiner triple system of order u , an STS (u) . We present several different constructions for designs of this form. As a consequence, we completely settle the extremal case v= 2u+ 1 , for which a list of possible exceptions had remained for close to 30 years. Our new constructions also provide the first infinite classes for the more general problem. We reduce the other maximal case v= 2u+ 3 to (at present) three possible exceptions. In addition, we obtain results for other cases of the form v= 2u+w and also near v= 3u . Our primary method introduces a new type of Kirkman frame which contains group divisible design subsystems. These subsystems can occur with different configurations, and we use two different varieties in our constructions.
期刊介绍：
The Journal of Combinatorial Designs is an international journal devoted to the timely publication of the most influential papers in the area of combinatorial design theory. All topics in design theory, and in which design theory has important applications, are covered, including:
block designs, t-designs, pairwise balanced designs and group divisible designs
Latin squares, quasigroups, and related algebras
computational methods in design theory
construction methods
applications in computer science, experimental design theory, and coding theory
graph decompositions, factorizations, and design-theoretic techniques in graph theory and extremal combinatorics
finite geometry and its relation with design theory.
algebraic aspects of design theory.
Researchers and scientists can depend on the Journal of Combinatorial Designs for the most recent developments in this rapidly growing field, and to provide a forum for both theoretical research and applications. All papers appearing in the Journal of Combinatorial Designs are carefully peer refereed.
京公网安备 11010802042870号
京ICP备2023020795号-1
分享
求助内容：
应助结果提醒方式：
扫码关注我们
