Extremal and Structural Aspects of Graph Minor Theory

这个提议的目标是系统地研究极小闭图类的极值和结构性质。图子式理论是图论中一个很深很丰富的领域，最初由Robertson和Seymour在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。作为理论的一部分而开发的一些方法已经成功地用于实际计算中。图子理论中的一个中心结果是Robertson和Seymour的图结构定理，它给出了不包含固定图作为子图的图的近似结构描述。PI建议继续他与Robin托马斯正在进行的长期联合项目，该项目的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接性的严格界限，保证某些未成年人和相关配置（链接，拓扑未成年人等）的存在。in large大graphs图形. ** PI还建议研究图子理论的极值方面。在这个方向上的建议的主要目标之一是表明，密度的每一个小封闭类的图是由图的有界路宽。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猜想是一个长期存在的公开问题，它极大地加强了四色定理。它可能是图论中最著名的开放问题。PI最近宣布了一个证明，与Zdenek Dvorak联合，对猜想的一个放松，改进了Kawarabayshi和Mohar，Wood，Liu和Oum的早期结果。PI建议将这一结果扩展到几个方向，特别是研究与自举渗流的有趣联系，这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.