Extremal and Structural Aspects of Graph Minor Theory

图小论的极值和结构方面

• 批准号: RGPIN-2017-05010
• 负责人: Norin, Sergey
• 金额: $ 1.46万
• 依托单位: McGill University
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2020
• 资助国家: 加拿大
• 起止时间: 2020-01-01 至 2021-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=711397
• 关键词: Extremal; Structural; Aspects; Graph; Minor

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.

该提案的目的是对少量封闭图类别的极端和结构特性进行系统研究。图形理论是图理论的深层理论，最初是由罗伯逊和西摩开发的，其中有23篇论文。它仍然是具有广泛算法应用的研究领域。作为理论的一部分开发的一些方法已成功地用于实际计算中。 图形理论的核心结果之一是罗伯逊和西摩的图形结构定理，该定理给出了不包含固定图作为小调的图的近似结构描述。 PI提议继续与罗宾·托马斯（Robin Thomas）继续他正在进行的长期联合项目，其目标是对该理论的许多方面的完善。特别是，该项目的目标之一是获得有关连通性的紧密界限，以确保某些未成年人的存在和相关配置（链接，拓扑未成年人等）在大图中。 PI还提出了图形理论的极端方面的研究。该提案朝这个方向的主要目标之一是表明，通过有界路径的图，可以实现每个少量封闭的图形类别的密度。第二个目标是计算特定的次要类别的密度，并为这种类型的问题开发通用工具。 最后，PI提议研究哈德威格猜想的放松。 Hadwiger的猜想是一个长期存在的开放问题，它极大地增强了四色定理。这可能是图理论中最著名的开放问题。 PI最近宣布了与Zdenek Dvorak的联合证明，可以放松猜想，并改善了Kawarabayshi和Mohar，Wood，Liu和Oum的早期结果。 PI建议将此结果扩展到多个方向，特别是研究了与Bootstrap Percolation的有趣联系，这是一个在概率组合和理论物理学中研究的概念。