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.

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。