Structural properties and generalizations of combinatorial designs

组合设计的起源在于它们在检验对象之间相互作用的统计实验设计中的应用。在对象以圆形方式排列的情况下，某些类型的实验可以通过图的循环分解来建模，也称为循环设计。此外，组合设计在软件测试、编码理论和通信系统等领域都有应用。*我建议研究循环设计的结构方面，包括这些设计的点和块的着色以及循环设计的交性质。设计点的着色在实验设计中有应用。设计的块着色与图的因子分解和著名的Oberwolfach问题有关。我的目标是构建某些类型的自行车设计与理想的着色性能。*此外，我还将进一步发展广义设计、包装和覆盖的理论。这些对象的某些情况下模拟了普通设计的各种颜色。此外，广义设计、填充和覆盖涉及许多现有的组合设计类型，因此它们的研究将为组合设计理论家提供有力的工具。*最后，我将考虑一类称为圈交图的图，它是基于圈设计中的圈的交而形成的。这些图的类似图以前已经被研究过用于其他类型的设计。将考虑的一个问题是，设计中的循环是否可以排序，以便连续的循环以特定方式相交。

The origin of combinatorial designs lies in their application to the design of statistical experiments to test interactions between objects. In cases where objects are to be arranged in a circular manner, certain types of experiments can be modelled by cycle decompositions of graphs, also called cycle designs. In addition, combinatorial designs have applications to areas such as software testing, coding theory and communication systems.****I propose to study structural aspects of cycle designs including colourings of the points and blocks of these designs and intersection properties of cycle designs. Colourings of the points of designs have applications in experimental design. Block colourings of designs relate to graph factorizations and the well-known Oberwolfach problem. I aim to construct certain types of cycle designs with desirable colouring properties. ****In addition, I will further the development of the theory of generalized designs, packings and coverings. Certain cases of these objects model various types of colourings of ordinary designs. Moreover, generalized designs, packings and coverings relate to many existing types of combinatorial designs, so their study will prove a powerful tool to combinatorial design theorists.****Finally, I will consider a class of graphs called cycle intersection graphs, which are formed based on the the intersection of cycles in a cycle design. Analogues of these graphs have previously been studied for other classes of designs. One question that will be considered is whether the cycles in a design can be ordered so that consecutive cycles intersect in a particular way.****