圈设计中的三个公开问题的研究

本项目拟研究组合设计中与圈设计相关的三个公开问题，即Oberwolfach问题、Hamilton-Waterloo问题和几乎可分解圈设计的存在性问题。以圈设计的研究为纽带，将这些问题的研究有机地结合起来，研究问题解的存在性。这些问题吸引了许多学者的关注并取得丰富的研究成果，成为组合设计的研究热点问题。虽然这些问题的研究成果很多，但是离整个问题的完全解决还很遥远，值得进一步探究。本项目拟通过代数、组合、概率等工具从构造方法角度创新圈设计的研究方法。比如运用Skolem序列、差矩阵等代数工具给出Oberwolfach问题解的构造方法，将α-标号等标号方法和圈支架等组合结构融入Hamilton-Waterloo问题解的构造方法，用概率方法研究几乎可分解圈设计的渐近存在性。申请人在这些问题的研究方面已取得一系列成果，积累了丰富的研究经验。项目的实施将推动这些问题的研究，丰富圈设计的研究成果。

This project mainly studies three open problems related to circle design in combinatorial design, namely, Oberwolfach problem, Hamilton-Waterloo problem and the existence of almost resolvable cycle system. With the research of circle design as a link, the research of these problems is organically combined to study the existence of solution. These problems have attracted the attention of many scholars and achieved rich research results, and become a hot issue in combinatorial design. Although there are many research results on these problems, it is still far away from the complete solution of the whole problem, which deserve further study. This project plans to apply algebra, combinatorics, probability and other tools to innovate the research method of circle design from the perspective of construction method. For example, the construction method of Oberwolfach problem solution is given by using the algebra tools such as Skolem sequence and difference matrix. Labeling methods and combinatorial structures are integrated into the construction method of Hamilton-Waterloo problem solution, such as α-labeling and cycle frame. The asymptotic existence of almost resolvable cycle system is studied by probability method. Its implementation will promote the research of these problems and enrich the research results of circle design.