Hadwiger's Conjecture and Ramsey Related Problems

本研究项目旨在研究结构图论和极值组合学领域中的各种基本问题。这类问题通常涉及其他领域，包括理论计算机科学、几何、信息论、调和分析和数论。在这些问题上的进展将促进我们对图论和组合学的相关方面的理解。PI和她的合作者最近在其中大多数方面取得了进展。预计在这些问题上的进一步工作将导致新的方法和应用。该项目研究与著名的Hadwiger猜想和Ramsey相关问题有关的问题。具体地说，PI和她的合作者计划探索以下问题：证明了在七个顶点上没有团的图是7-可染的；推广了没有团的图的Mader界；研究了独立数为2的图的Hadwiger猜想的极小反例；研究了具有小Colin de Verdiere参数的图的极值函数；确定了偶圈、轮图和完全图的Gallai-Ramsey数的精确值；估计了上临界图的最小边数。在这些问题上的进展无疑将导致开发以前遥不可及的新方法和途径，并可能允许进一步令人兴奋的发展。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

This research project aims to study a variety of fundamental problems in the areas of structural graph theory and extremal combinatorics. Often such problems are related to other areas including theoretical computer science, geometry, information theory, harmonic analysis and number theory. Progress on these problems will advance our understanding of related aspects of graph theory and combinatorics. The PI and her collaborators have recently made progress on most of them. It is expected that further work on these problems will lead to new methods and applications. The project investigates problems related to the well-known Hadwiger's conjecture and Ramsey related problems. Specifically the PI and her collaborators plan to explore the following problems: proving every graph with no clique minor on seven vertices is 7-colorable; generalizing Mader's bound for graphs with no clique minors; studying the minimal counterexamples to Hadwiger's conjecture for graphs with independence number two; and studying the extremal function for graphs with small Colin de Verdiere parameter; determining the exact values of Gallai-Ramsey numbers of even cycles, wheels, and complete graphs; and estimating the minimum number of edges of co-critical graphs. Progress on these problems will undoubtedly lead to the development of new methods and approaches that were too far out of reach before, and are likely to allow further exciting developments.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.