Structure in Designs, Coverings and Decompositions

这份申请是为了更新我的发现补助金，这将有助于正在进行的研究活动。它将有助于支持人力资源规划署，促进合作项目，并加强相关成果的传播。组合设计和相关的图分解和因式分解，以及包装和覆盖，提供了一个理想的方式来理解复杂的离散结构，如网络的相互作用特性，一种方法来研究这种结构的“互连”属性。拟议的研究计划的中心焦点是组合设计和相关对象的结构的调查，并考虑具有特定结构的设计，或缺乏。在这一领域的发展将提供一个更深入的了解所涉及的对象的结构，以及洞察到其他组合问题。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，还有潜在的应用程序，如算法效率，软件和网络测试等各种问题。** 我长期以来一直致力于寻找各种因子分解，最初是一致和类一致可分解的设计，并将这项工作扩展到包括更一般的因子分解，最著名的是循环因子分解。我还考虑了不可能进行因式分解的情况。最近，我对覆盖阵列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。这也将为HQP在这些领域的培训提供充足的机会。* 长期存在的Oberwolfach问题，由Ringel在20世纪60年代提出，多年来一直受到广泛关注，最近取得了一些进展。相关的汉密尔顿滑铁卢问题，在那里我有重大的成功，需要一个因式分解的完整的图形到各种循环类型。这个项目的目标之一是建立在我最近的结果，并继续调查这些和其他相关的问题。另一个目标是考虑组合对象不能有一个特定的结构，如resolvability，或有多余的结构，如DoublyResolvable设计 ** 覆盖阵列已经收到了很大的兴趣，最近由于其在测试中的应用，特别是软件和网络测试。本提案的目标之一是进一步研究这些对象并进一步概括它们。这包括具有受限交互集的情况以及对序列数组或传递数组的概括。**

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. **