图论中的整数流与圆流

项目名称： 图论中的整数流与圆流

项目编号： No.11501504

项目类型： 青年科学基金项目

立项/批准年度： 2016

项目学科： 数理科学和化学

项目作者： 吴叶舟

作者单位： 浙江大学

项目金额： 18万元

中文摘要： 本项目旨在研究图论中的关于流理论方面的内容，包括整数流和圆流相关问题。由Tutte提出的3-流猜想和5-流猜想是图论中的两个核心难题。Jeager进一步将它们推广为统一的圆流猜想。同时考虑到问题的难度，Jeager还提出了它们的弱化版本: 弱3-流猜想和弱圆流猜想。即便如此，这两个弱化的猜想长期悬而未决。直到2011年，弱3-流猜想和弱圆流猜想才最终被丹麦科学院院士Carsten Thomassen解决。随后本项目申请人和博士导师张存铨教授在与Laszlo Miklos Lovasz及Carsten Thomassen的合作中，进一步改进了这两个结果，并在与Dong Ye及Wenan Zang的合作中将Thomassen的弱3-流定理推广到符号图上。本项目将继续研究关于图的整数流和圆流的相关研究, 重点是图与符号图中的3-流问题，拟完全解决3-流猜。

中文关键词： 整数流；符号图；群连通；3-流猜想；圆流猜想

英文摘要： In this project, we investigate integer flow and circular flow problems on graphs. Tutte's 3-Flow Conjecture and 5-Flow Conjecture are two kernel problems of graph theory. Jeager generalized those two conjectures to a stronger one, known as the Circular Flow Conjecture. Considering the hardness of the problems, Jeager also proposed the weaker versions of the Conjectures, called weak 3-Flow Conjecture and weak Circular Flow Conjecture. Howerver even the two weaker conjectures were still remained open in a long dormant period. In 2011, Carsten Thomassen (member of the Royal Danish Academy of Sciences and letters) finally solved the weak 3-Flow Conjecture and weak Circular Flow Conjecture. Later in cooperation with Laszlo Miklos Lovasz and Carsten Thomassen, my PhD advisor professor Chunquan Zhang and I improved those two results, and in cooperation with Dong Ye and Wenan Zang, we generalized Thomassen’s weak 3-Flow Theorem to signed graphs. In the future we shall continue study the problems related to integer flows and circular flow, focusing on 3-flow probem on graphs and signed graph, and try to solve 3-Flow Conjecture completely.

英文关键词： integer flow;signed graph;group connectivity;3-Flow Conjecture;Circular Flow Conjecture