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。这个领域与计算机科学、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、社交网络、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信道分配/分配的有效模型，而流问题最初出现在业务或网络优化中。这个项目涉及网络中的结构问题。 Tutte提出的图/地图着色问题的对偶问题--图流理论是本项目的主要研究对象。最近，首席研究员（与Lovasz，Wuassen和Wu合作）成功地证明了每个6边连通图都允许无零3流，并且（与Wuassen和WVU的团队合作）发现了强连通方向和循环流指数之间的关系。首席研究员将继续他的研究工作，在这个方向上，通过确定流动指数的图形与各种连接，调查流动的图形嵌入在非定向表面，研究模方向的连接较少的图形更好的解决方案，以图特的流动aesthetures。

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.