Extremal and Structural Aspects of Graph Minor Theory

这一建议的目的是系统地研究次闭图类的极值和结构性质。图子论是图论中一个深刻而丰富的领域，最初是由Robertson和Seymour在一系列23篇论文中发展起来的。它仍然是一个活跃的研究领域，有着广泛的算法应用。作为该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。*图次要理论的核心结果之一是Robertson和Seymour的图结构定理，它给出了不包含固定次要图的图的近似结构描述。PI提议继续他与罗宾·托马斯正在进行的长期联合项目，其目标是对这一理论的许多方面进行完善。特别是，该项目的目标之一是获得连接的严格界限，以保证存在某些次要项和相关配置(链接、拓扑次要项等)。在大图中。*PI还建议研究图次要理论的极端方面。这个方向的建议的主要目的之一是证明每一个次闭图类的密度是由有界路宽度的图获得的。第二个目标是计算特定次闭类的密度，并为这类问题开发通用工具。*最后，PI建议研究Hadwiger猜想的松弛。Hadwiger猜想是一个长期悬而未决的问题，它极大地加强了四色定理。这可能是图论中最著名的公开问题。PI最近与Zdenek Dvorak联合宣布了一个证明，证明了猜想的一个松弛，改进了Kawa abayshi和Mohar，Wood以及Liu和Oum的早期结果。PI建议在几个方向上扩展这一结果，特别是调查与Bootstrap渗流的有趣联系，Bootstrap渗流是概率组合学和理论物理学中研究的一个概念。

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.