Structure in Designs, Coverings and Decompositions

这份申请是为了续期我的发现基金，这将促进正在进行的研究活动。它将起到支持总部基地计划、促进合作项目和加强传播相关成果的作用。 组合设计和相关的图分解和因式分解，以及包装和覆盖，为理解复杂离散结构(如网络)的相互作用性质提供了一种理想的方法，也提供了一种研究这种结构的互连性的方法。建议的研究计划的中心焦点是调查组合设计和相关对象的结构，并考虑具有特定结构的设计或没有特定结构的设计。这一领域的发展将提供对所涉及对象的结构的更深入的理解，以及对其他组合问题的洞察。设计在统计学、编码理论和调度方面有着众所周知的应用。此外，在算法效率、软件和网络测试等问题上也有潜在的应用。 我长期以来一直致力于寻找各种分解，最初是一致和类一致可分解的设计，将这项工作扩大到包括更一般的分解，最著名的是循环分解。我还考虑过不可能进行因式分解的情况。最近，我对覆盖数组的相关领域产生了兴趣。这个项目将支持我在所有这些领域的工作。该提案将使用最先进的组合技术以及先进的算法方法来研究这些结构。它还将为HQP在这些领域和其他领域的培训提供充足的机会。 Oberwolfach问题由来已久，由Ringel在20世纪60年代引入，多年来受到了极大的关注，最近取得了一些进展。相关的汉密尔顿滑铁卢问题，我在那里取得了巨大的成功，需要将完整的图因式分解成各种循环类型。这个项目的目标之一是在我最近成果的基础上，继续调查这些问题和其他相关问题。另一个目标是考虑不具有特定结构(例如可分解性)或具有过剩结构(例如双重可分解设计)的组合对象 由于覆盖阵列在测试中的应用，特别是在软件和网络测试中的应用，最近引起了人们的极大兴趣。这项提案的目标之一是进一步研究这些对象及其进一步的概括。这包括具有受限交互集和序列数组或传递数组的泛化的情况。

This application is for the renewal of my discovery grant which will facilitate ongoing research activities. It will serve to support HQP, promote collaborative projects and enhance the dissemination of related results. Combinatorial designs and related graph decompositions and factorizations, as well as packings and coverings, provide an ideal way to understand the interaction properties of complex discrete structures, such as networks, a way to investigate the 'interconnectedness' properties of such structures. The central focus of the proposed research program is the investigation of the structure of combinatorial designs and related objects and to consider designs with a particular structure, or lack thereof. Development in this area will provide a deeper understanding of the structure of the objects involved as well as insight into other combinatorial questions. Designs have well-known applications to statistics, coding theory and scheduling. In addition, there are potential applications to such diverse questions as algorithmic efficiency, software and network testing. I have long been involved with finding various kinds of factorizations, initially uniformly and class-uniformly resolvable designs, broadening this work to include more general factorizations, most notably cycle factorizations. I have also considered cases where factorization is not possible. More recently, I have become interested in the related area of Covering Arrays. This project will support my work in all of these areas. The proposal will use state of the art combinatorial techniques as well advanced algorithmic methods to investigate these structures. It will also provide ample opportunity for HQP training in these areas an beyond. The longstanding Oberwolfach problem, introduced by Ringel in the 1960s has received much attention over the years, with several recent advances. The related Hamilton Waterloo problem, where I have had significant success, requires a factorization of the complete graph into a variety of cycle types. One of the objectives of this project is to build on my recent results and continue investigating these and other related problems. Another goal is to undertake a consideration of combinatorial objects which fail to have a particular structure, such as resolvability, or have excess structure, such as Doubly Resolvable Designs Covering arrays have received much interest of late due to their applications in testing, particularly software and network testing. One of the goals of this proposal is to further investigate these objects and further generalisations of them. This includes cases with restricted interaction sets and generalisations to sequence arrays or transitive arrays.