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，特别是图中的流动、着色和连通性。该领域与计算机科学、运筹学、数据挖掘和神经科学密切相关，其中图是用于建模网络的抽象数学概念，例如通信和运输网络、社会网络、监测数据、生物信息学路径和神经网络。各种图形着色问题被认为是无线电信道分配/分配的有效模型，而流量问题最初出现在优化交通或网络中。这个项目是关于网络结构问题的。由Tutte引入的整数流理论是图/图着色问题的对偶，是本项目的主要课题。最近，首席研究员（与Lovasz， Thomassen和Wu合作）成功地证明了每个6边连通图都承认无零的3流，并且（与Thomassen和WVU的团队合作）发现了强连接方向和循环流动指数之间的关系。首席研究员将在这个方向上继续他的研究工作，通过确定具有各种连通性的图的流动指标，研究嵌入在不可定向表面上的图的流动，研究较少连接图的模方向，以更好地解决Tutte的流动猜想。

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.