边染色图与有向图中的几类极值问题

There are abundent contents in the study of edge-colored graphs and digraphs, particularly extremal problems are very important in structural graph theory, extremal graph theory and Ramsey theory. In this project, we mainly focus on rainbow Turan problems (the existence of rainbow matchings and rainbow cycles) in edge-colored graphs, the partition of digraphs proposed by Alon and Stiebitz, the existence of directed paths and directed cycles (Thomassé Conjecture and Jackson Conjecture) and antimagic labelings of directed graphs. We are willing to impove conditions of existence of rainbow matchings in edge-colored (hyper) graphs, explore the relationship between directed graphs and edge-colored graphs and try to solve several problems and conjectures. The study of this project includes extremal graph theory, digraph theory, combinatorial design and matroid theory, which will be very significant not only for the development but also for the innovation of graph theory.

边染色图与有向图中蕴含着丰富的研究内容，其中的极值问题是结构图论、极值图论和Ramsey理论的研究重点。本课题主要研究边染色图中彩色图论问题（彩色匹配、彩色圈的存在性），Alon与Stiebitz提出的有向图的划分问题、有向路和圈的存在性（Thomassé猜想和Jackson猜想）以及有向图的反魔幻标号等。我们将改进边染色图（超图）中的彩色匹配的存在性条件，挖掘有向图与边染色图之间的联系，力求解决几个难题和猜想。 本课题的研究涉及到极值图论，有向图理论，组合设计，拟阵理论等多个领域，问题的解决对于图论的理论发展和创新都有非常重要的意义。

边染色图与有向图中蕴含着丰富的研究内容，其中的极值问题是结构图论、极值图论和Ramsey理论的研究重点。我们主要研究了图的局部反魔幻标号问题，边染色图中两类极值问题，以及图系统中彩色哈密顿圈问题和拟随机超图中的子图问题等。利用组合数学领域近年来非常重要的吸收引理以及超图上的强正则引理等方法，渐进解决了Aharoni等人提出的有关图系统中彩色哈密顿存在性的猜想，彻底回答了Mubayi教授等人提出的拟随机超图中哈密顿圈存在性的公开问题。本课题的研究涉及到极值图论，有向图理论，组合设计，拟阵理论等多个领域，问题的解决对于图论的理论发展和创新都有非常重要的意义。