度和与圈相关的一些子图结构

子图结构是图论的重要研究领域，而度和与子图结构是结构图论的经典研究方向之一。它研究的是当限定阶数的独立集的某个子集的度和较大(即局部结构相对稠密)时某个特定子结构的存在性问题。diff(G)、不交圈和圈划分问题是Hamilton理论的重要研究内容，而图的不交圈与圈划分问题一直是度和与子图结构的研究热点。本项目旨在研究度和与diff(G)、不交圈和圈划分的相关问题，拟研究的内容是：从最长圈的角度出发，利用圈结构分析，解决或部分解决Liu-Lu-Tian猜想；在Fan-型条件下，研究无向图的圈划分问题，推广Brandt等人关于Ore条件下圈划分的结果；对于Chiba和Yamashita提出的问题，即Bermond-Thomassen猜想的推广形式，从包含完美匹配的均衡二部图出发，研究k比较小的情况。

Structures of subgraphs is an important field of graph theory, and investigating degree sums and structures of subgraphs is a classical branch of structure graph theory. It states that if the degree sums of some subset of a given independent set is large, that is, the local structure of a graph is not so sparse, then it will necessarily contain an orderly substructure. The relative length, disjoint cycles and partitions into cycles are important research contents of Hamilton theory, and the problems of disjoint cycles and partitions into cycles have been a research focus in graph theory. This project aims at studying some frontier problems of the relative length, disjoint cycles and partitions into cycles, and the research contents of this project are as follows: using the method of extending cycle from the perspective of longest cycles, solving or partially solving Liu-Lu-Tian Conjecture; studying partitions into cycles under Fan-type condition, generalizing the result of Brandt et al. on Ore`s condition; investigating disjoint cycles of balanced bipartite graphs with perfect matching under degree sums for small k, which will help us to study the generalization of Bermond-Tomassen Conjecture posed by Chiba and Yamashita.