Research on Graph Coloring and Graph Structure

本研究项目研究图中某些构型的存在，以及这些构型与图的整体性质之间的关系。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，每个集合中没有边。这个项目寻求其他配置的类似结果，以及保证图中存在某些配置(如圈和匹配)的充分条件。本课题还研究了适合研究生的图的圈和超图的匹配问题。本课题研究了图的着色和图的结构两个问题。Hajos猜想不包含5点完全图的剖分的图是4-可染的，从而推广了四色定理。关于这一猜想的前人工作使人们对这些图的结构有了实质性的了解，以及它与几个与图结构有关的重要问题的联系，包括关于非分离路的Lovasz猜想和关于3-链图的Thomassen猜想。另一个问题涉及将不包含给定树的图的色数定为导出子图。将考虑使用特殊的树木来洞察这一问题。这一奖项反映了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 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.