Structural properties and generalizations of combinatorial designs

组合设计的起源在于将其应用于统计实验的设计中，以测试物体之间的相互作用。在以循环方式安排对象的情况下，某些类型的实验可以通过图形的循环分解来建模，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也有应用。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叉特性。设计点的着色在实验设计中有应用。设计的块着色与图分解和著名的Oberwolfach问题有关。我的目标是构建具有理想着色性能的某些类型的循环设计。此外，我将进一步发展理论的广义设计，包装和覆盖。这些对象的某些情况下模型的各种类型的普通设计的颜色。此外，广义设计、包装和覆盖与许多现有的组合设计类型有关，因此他们的研究将证明是组合设计理论家的有力工具。最后，我将考虑一类称为循环相交图的图，它是基于循环设计中循环的相交而形成的。这些图的类似物以前已经研究过其他类型的设计。要考虑的一个问题是，设计中的循环是否可以有序，以便连续的循环以特定的方式相交。

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.