Extremal and Structural Aspects of Graph Minor Theory

本文的目的是系统地研究图的小闭类的极值性质和结构性质。图小理论是图论中一个深刻而丰富的领域，最初由Robertson和Seymour在一系列23篇论文中发展起来。它仍然是一个活跃的研究领域，具有广泛的算法应用。作为理论的一部分发展起来的一些方法已经成功地应用于实际计算中。******图的次要理论的中心结果之一是Robertson和Seymour的图结构定理，它给出了不包含固定图作为次要的图的近似结构描述。PI建议继续他与Robin Thomas正在进行的长期联合项目，其目标是改进该理论的许多方面。特别是，该项目的目标之一是获得连通性的紧界，以保证在大图中存在某些次要和相关构型（连杆，拓扑次要等）。****** PI还提出了图小理论的极值方面的研究。在这个方向上提出的主要目标之一是证明每一个小闭类图的密度是由有界路径宽度的图获得的。第二个目标是计算特定小闭类的密度，并为这类问题开发通用工具。******最后，PI提出研究哈维格猜想的松弛性。哈维格猜想是一个长期存在的开放问题，它极大地强化了四色定理。它可能是图论中最著名的开放问题。PI最近宣布了与Zdenek Dvorak合作的一项证明，证明了该猜想的一个松弛，改进了Kawarabayshi、Mohar、Wood、Liu和Oum的早期结果。PI建议在几个方向上扩展这一结果，特别是研究与自举渗透的有趣联系，这是一个在概率组合学和理论物理学中研究的概念。

The goal of this proposal is systematic investigation of extremal and structural properties of minor-closed classes of graphs. Graph minor theory is a deep and rich area of graph theory, initially developed by Robertson and Seymour in a series of twenty three papers. It continues to be an active area of research with extensive algorithmic applications. Some of the methods developed as part of the theory have been successfully used in practical computations.******One of the central results in graph minor theory is the graph structure theorem of Robertson and Seymour, which gives an approximate structural description of graphs that do not contain a fixed graph as a minor. The PI proposes to continue his ongoing long term joint project with Robin Thomas, the goal of which is a refinement of many aspects of this theory. In particular, one of the goals of the project is to obtain tight bounds on connectivity which guarantees existence of certain minors and related configurations (linkages, topological minors, etc.) in large graphs. ******The PI also proposes investigation of extremal aspects of graph minor theory. One of the main goal of the proposal in this direction is to show that the density of every minor-closed class of graphs is attained by graphs of bounded pathwidth. The second goal is to compute density of particular minor-closed classes and develop generic tools for this type of problems.******Finally, the PI proposes to investigate relaxations of Hadwiger's conjecture. Hadwiger's conjecture is a longstanding open problem, which greatly strengthens the four-color theorem. It is possibly the most famous open problem in graph theory. The PI has recently announced a proof, joint with Zdenek Dvorak, of one relaxation of the conjecture, improving on earlier results of Kawarabayshi and Mohar, Wood, and Liu and Oum. The PI proposes to extend this result in several directions, in particular, investigating intriguing connections with bootstrap percolation, a concept investigated in probabilistic combinatorics and theoretical physics.