图中与森林相关的若干极值问题研究

基于图参数的极值问题是图论科学研究中的热点问题之一。本项目主要探讨图中与森林相关的几类极值问题，拟解决其中部分富有挑战性的未解决问题。研究内容主要包括：(1)图中过线性森林的最长圈问题。我们将利用与最长圈C以及G-C的分图密切关联的“好集合”、“辅助二部图”等新概念和新技巧，研究图G中过特定元素的最长圈长度。该部分的研究将推进长期悬而未决的Locke-Zhang问题的解决；(2)森林的饱和数问题，该问题是著名的Turán问题对应的最小边数问题。我们拟利用Berge-Tutte公式、点移动、子图扩张等技巧和方法,研究森林的饱和数。我们的研究有望推动极值图论等相关数学领域的进展，为管理科学及理论计算机科学等提供一些有意义的数学依据。

Extremal problem based on some graphic parameters is one of the hot problems in graph theory．In this proposal, we mainly study the extremal problems which are related to forest, and we hope to solve some challenging problems in extremal graph theory. The main content of this proposal contains the following： Firstly, we mainly study the length of long cycles passing through linear forest. By employing such new concepts and techniques as“good sets”and“auxiliary bipartite graphs”that are closely related with a longest cycle C of G and components of G-C, we will investigate the length of a longest cycle passing through some given elements in a graph. As a result, the project will contribute to the earlier solution to the Locke-Zhang open problem. Secondly, we study saturation number for some forest, which is the minimum number of edges corresponding to the famous Turán problem. By introducing the methods and techniques such as Berge-Tutte formula, moving vertex，expansion of the subgraph, we investigate the saturation number of forests. The research of this proposal is expected to promote the development of mathematical fields such as extremal graph theory, and provide some meaningful mathematical basis for management science and theoretical computer science.