图与超图分解及谱形式极值问题

图或超图的H-分解问题和Turán型极值问题关系非常紧密，都是极值图论研究中的基本问题，而极值问题谱形式为Turán型极值问题研究提供了新的思路和角度。本项目主要围绕图的H-分解的Pikhurko-Sousa猜想（当H色数大于等于3时，n阶图的H-分解的极图边数等于以H为限制子图的n阶极图边数），超图的H-分解问题和极值问题谱形式开展研究。希望证明Pikhurko-Sousa猜想当H为点临界图时成立，进而完全证明该猜想；解决H为退化情形时超图的H-分解问题并探索H为非退化情形时超图H-分解的一般公式；解决Nikiforov提出的有关极值问题谱形式的两个猜想。这些问题的解决和取得进展必然会促进极值图论理论的完善和发展。

The problem of H-decomposition of graphs or hypergraphs has tight relationship with Turán type extremal problems, both of them are fundamental problems in extremal graph theory, the study of the spectral extremal graph theory provides new ideas for the study of the corresponding Turán type extremal problems. The project mainly concerns the conjecture of Pikhurko-Sousa related to H-decomposition of graphs, the H-decomposition of hypergraphs and the spectral versions of some extremal problems. We want to solve the conjecture of Pikhurko-Sousa when H is vertex-critical and then solve the conjecture completely, solve the H-decomposition problem when H is degenerate and find the general function for the H-decomposition of hypergrapgs when H is nondegenerate, solve two conjectures related to the spectral extremal graph theory proposed by Nikiforov. The solvement of those problems will improve the developement of the extremal graph theory.