Designs and cycle decompositions

图可以被认为是网络的数学表示;它由一组顶点（通常表示为点）和表示顶点之间关系的边（连接顶点对的线）组成。假设在一个图中，我们从一个顶点v开始，遍历一个边序列，每个边都与前一个边相邻。如果我们回到v，沿途沿着没有重复任何顶点，我们就遍历了一个循环，其长度等于遍历的边的数目。图的圈分解是将图的边集划分为圈。高度结构化图的循环分解，例如完全图（每对顶点都有一条边连接）和完全多部图（顶点出现在部分中，任何对之间的边都在不同部分中），可以被认为是一类组合设计，并在某些类型的统计实验设计中有应用。 例如，考虑D. H. Rees（Some designs of use in serology，Biometrics 23（1967），779-791）.一位生物学家正在使用一种称为Ouchterlony凝胶扩散技术的方法进行一项涉及v型病毒的测试。 她可以在培养皿中将k病毒样本排列成圆形围绕抗体。当病毒和抗体反应时，沉淀物形成，相邻样品之间沉淀物的形状表明抗体是否对两者的共同成分产生反应。出于这个原因，她想设计她的实验，使每对病毒彼此相邻出现一次;将v个顶点上的完整图分解为长度为k的循环允许她这样做。如果病毒出现在家族中，其中家族的不同成员不需要比较，则完全多部图的分解将是适当的。这个建议将考虑与某些图族的循环分解相关的存在性问题，例如完全多部图和完全有向图。此外，将探讨与顶点或循环着色相关的结构属性的分解，以及与此类分解相关的图的邻接属性。 拟议的研究将代表该领域知识的进步，提案中描述的主题与调度，代码和通信等领域的应用有关。该研究将提供本科生、研究生和博士后培训机会。

A graph can be thought of as a mathematical representation of a network; it consists of a collection of vertices (often represented as points) together with edges (lines joining pairs of vertices) representing relationships between vertices. Suppose that in a graph, we start at a vertex v and traverse a sequence of edges, each adjacent to the previous. If we arrive back at v without repeating any vertex along the way, we have traversed a cycle, whose length equals the number of edges traversed. A cycle decomposition of a graph is a partition of its edge set into cycles. Cycle decompositions of highly structured graphs, such as complete graphs (which have an edge joining each pair of vertices) and complete multipartite graphs (whose vertices occur in parts, with edges between any pair in different parts), can be considered as a class of combinatorial design, and have applications in the design of certain types of statistical experiment.  Consider, for instance, the following example described in a paper of D.H. Rees (Some designs of use in serology, Biometrics 23 (1967), 779-791). A biologist is conducting a test involving v types of virus using a method called the Ouchterlony gel diffusion technique.  She can arrange k virus samples in a circular shape around an antibody in a Petri dish. Precipitates form when a virus and antibody react, and the shape of a precipitate between neighbouring samples indicates whether the antibody is responding to a common component of the two. For this reason, she wants to design her experiment so that each pair of viruses appear next to each other once; a decomposition of the complete graph on v vertices into cycles of length k allows her to do this. If the viruses occur in families, where different members of a family are not to be compared, a decomposition of a complete multipartite graph would be appropriate. This proposal will consider existence questions related to cycle decompositions of certain families of graphs such as complete multipartite graphs and complete directed graphs. Additionally, decompositions with structural properties related to colouring of vertices or cycles will be explored, as will adjacency properties of graphs related to such decompositions. The proposed research will represent an advancement of knowledge in the area and the topics described in the proposal have connections to applications in areas such as scheduling, codes and communications. The research will provide opportunity for training at the undergraduate, graduate and postdoctoral levels.