子图存在性是结构图论和极值图论的中心课题之一，而图的长圈和哈密顿圈是子图存在领域的核心。本项目一方面以Dirac定理和Erdős-Gallai定理及其推广为主线，主要利用Bondy引理和Hopping引理等经典工具，结合局部最长圈的闭包理论和强支集理论，深入研究Woodall在1975年提出的一个长圈猜想和正则图中哈密顿圈存在性的Bollobás-Häggkvist问题的完整解，并重点研究其相应圈长结果的稳定性问题，寻找相应的谱类似型定理。另一方面则是研究范更华在1990年证明的指定顶点长路存在性定理的稳定性版本，并研究其在谱极值图论中的相关应用。本项目的研究是关于结构图论、极值图论和谱图论的结合，研究中注重结构分析、极图刻画和其他现代图论方法的综合运用。

The existence of subgraphs is a central topic of structural graph theory and extremal graph theory, and long cycles and Hamilton cycles are very important objects for research. In this project, we first focus on Dirac's theorem and Erdős-Gallai theorem and their generalizations. With the help of Bondy Lemma and Hopping Lemma, the closure theory of local maximal cycles and the theory of strong attachments of cycles, we shall deeply study a conjecture of Woodall proposed in 1975 and the Bolllobás-Häggkvikt problem on Hamilton cycles in regular graphs. We shall mainly solve the stability versions of each problem, and obtain the spectral analogs. On the other hand, we shall study the stability version of a theorem of Fan on long paths with specified end-vertices in 1990, and use it as a basic tool to solve open problems in spectral graph theory. We pay attention to analyze the structure of graphs and characterize the extremal graphs, and use other modern tools of graph theory.