图与超图镶嵌、覆盖及覆盖条件下极值问题研究

极值图论是组合图论中一个重要研究分支，在沃尔夫奖获得者Erdos及阿贝尔奖获得者Szemeredi为首的数学家们推动下得到快速发展并由此产生了很多新的理论和方法. 本项目主要围绕图与超图的镶嵌(tiling)、覆盖(covering)和覆盖条件下的极值问题开展研究，研究内容包括2014年国际数学家大会特邀报告人Kuhn和Osthus提出的有关图镶嵌的公开问题，Mycroft提出的关于k部k一致超图镶嵌问题的猜想，Falgas-Ravry和Zhao提出的超图覆盖的若干公开题及Chakraborti和Loh提出的覆盖条件下的若干极值问题等. 所用研究方法涵盖了极值图论、结构图论、组合设计、概率论和抽象代数等领域. 预期能够完全解决其中2-4个问题，在其它问题上取得重要进展并发展新的处理极值问题的方法和技巧. 通过对这些问题的研究阐明这些问题的发展规律及内在联系，进一步促进极值图论的发展.

Extremal graph theory is an important field in combinatorics and graph theory, under the leading of Erdos (Wolf prize) and Szemeredi (Abel prize), the researchers made a quick development in extremal graph theory and developed many new theories and methods. The project mainly concerns about tiling of graphs and hypergraphs, including some open problems of graph titlings proposed by Kuhn and Osthus (both of them are invited lectures at the International Congress of Mathematicians, 2014)，the conjecture of Mycroft about tiling k-partite k-uniform hypergraphs，some open problems of graph coverings proposed by Falgas-Ravry and Zhao, and som extremal problems under covering condition proposed by Chakraborti and Loh. The methods of the study include the methods used in extremal graph theory, structural graph theory, combinatorial design, probabilistic theory and abstract algebra. We hope that we can completely resolve 2-4 related problems, make progress on the other problems and develope new methods and techniques. Through the study of these problems, we can find the law of development and inner relations in these problems and improve the development of extremal graph theory.