Research on Graph Coloring and Graph Structure

本课题研究图中某些组态的存在性，以及这些组态与图的全局属性之间的联系。一个著名的例子是四色定理，该定理指出，如果一个图不包含两个特定的构型，那么它是平面的，并且它的顶点最多可以划分为四个集合，每个集合内没有边。该项目寻求其他构型的类似结果，以及保证图中某些构型（如循环和匹配）存在的充分条件。本课题还包含了适合研究生研究的关于图中的循环和超图中的匹配的问题。本课题主要研究图的着色和图的结构两个问题。Hajos推测不包含5顶点完全图的细分的图是4色的，这将推广四色定理。先前对这一猜想的研究使我们对这些图的结构有了实质性的了解，并将其与图结构的几个重要问题联系起来，包括Lovasz关于非分离路径的猜想和Thomassen关于3链图的猜想。另一个问题涉及到不包含给定树的图作为诱导子图的色数边界。我们将考虑特殊的树来深入了解这个问题。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值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 studies the existence of certain configurations in graphs, as well as connections between those configurations and global properties of graphs. A well-known example is the Four-Color Theorem, which states that if a graph does not contain two specific configurations, then it is planar, and its vertices can be partitioned into at most four sets with no edges within each set. This project seeks similar results for other configurations, as well as sufficient conditions that guarantee the existence of certain configurations (such as cycles and matchings) in graphs. This project also contains research problems related to cycles in graphs and matchings in hypergraphs that are suitable for graduate students.This research project investigates two problems on graph coloring and graph structure. Hajos conjectured that graphs containing no subdivision of the 5-vertex complete graph are 4-colorable, which would generalize the Four-Color Theorem. Prior work on this conjecture led to the substantial understanding of the structure of those graphs, as well as its connections to several important problems concerning graph structures, including Lovasz's conjecture on non-separating paths and Thomassen's conjecture on 3-linked graphs. Another problem involves bounding the chromatic number of graphs not containing a given tree as an induced subgraph. Special trees will be considered to gain insight into this problem.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.