Integer flows, circular flow indices and modulo orientations

该奖项支持首席研究员在图论方面的研究，特别是图中的流、着色和连通性。这一领域与计算机科学、运筹学、数据挖掘和神经科学密切相关，其中图是用于对网络建模的抽象数学概念，如通信和运输网络、社会网络、监视数据、生物信息学中的路径和神经网络。各种图着色问题被认为是无线电信道分配/分配的有效模型，而流问题最初是在优化交通或网络时出现的。该项目涉及网络中的结构问题。整数流理论是Tutte提出的图/图着色问题的对偶问题，它是本项目的主要主题。最近，主要研究者(与Lovasz，Thomassen和Wu合作)成功地证明了每个6-边连通图都有一个无处为零的3-流，并且(与Thomassen和西弗吉尼亚大学的团队合作)发现了强连通方向和圆流指数之间的关系。主要研究人员将通过确定具有各种连通性的图的流指数，研究嵌入在不可定向曲面上的图的流，研究连通度较低的图的模方向来更好地解决Tutte流猜想，从而继续他的研究工作。

This award supports the principal investigator's research in graph theory, and specifically flows, coloring, and connectivity in graphs. This area is closely related to computer science, operations research, data mining, and the neurosciences, where graphs are abstract mathematical notions used to model networks, such as communication and transportation networks, social networks, surveillance data, pathways in bioinformatics, and neural-networks. Various graph coloring problems have been considered as effective models for radio channel assignment/distribution, and flow problems originally arise in optimizing traffic or network. This project is concerned with structural problems in networks. Integer flow theory, introduced by Tutte as a dual of the graph/map coloring problem, is the major subject of this proposed project. Recently, the principal investigator (collaborating with Lovasz, Thomassen and Wu) successfully proved that every 6-edge-connected graph admits a nowhere-zero 3-flow, and (collaborating with Thomassen and the team at WVU) discovered a relation between strongly connected orientations and circular flow indices. The principal investigator will continue his research work in this direction by determining the flow indices for graphs with various connectivity, investigating flows of graphs embedded on non-orientable surfaces, studying modulo orientations of less connected graphs for better solutions to Tutte's flow conjectures.