符号图的流

The theory of integer flow, a classical field of graph theory, was proposed by Tutte as a tool to attack the four color problem. The graph containing positive edges and negative edges is called the signed graph, which can be used to describe more general model. In an orientable surface, vertex color of a graph is equivalent to the integer flow on its dual graph. However, in a nonorientable surface, dual problem of vertex color is the flow on signed graph.. The project would discuss the basic property of signed flow; characterize the relations between signed integer flow and signed modulo flow; enlarge the methods dealing with signed flow; determine the minimum values of signed flows on the graphs with special structure, such as, 3-edge connected graphs, Hamiltonian graphs; get primary results concerning signed 6-flow conjecture and signed 3-flow conjecture, attempt to solve these two conjectures.. This project should enrich the theory of integer flow, and promote its development.

整数流理论最早是由Tutte提出, 用于研究四色问题, 随后成为图论的经典研究方向. 符号图是即含正边又含负边的图, 是一般图模型的推广. 在可定向曲面上, 顶点染色的对偶问题是一般整数流；在不可定向曲面上, 顶点染色的对偶问题是符号图的整数流. . 本项目主要旨在研究符号流的基本性质, 刻画符号模流与符号整数流之间的关系, 拓展符号流的研究方法; 确定各种具有特殊结构的图类中符号流的存在性, 如3-边连通符号图, Hamilton符号图等; 针对符号3-流猜想和符号6-流猜想取得阶段性成果, 试图证明或逼近这两个猜想.. 本项目的顺利实施将拓展整数流的研究内容, 推动该领域的发展.

整数流理论最早是由Tutte提出, 用于研究四色问题, 随后成为图论的经典研究方向. 符号图是即含正边又含负边的图, 是一般图模型的推广. 在可定向曲面上, 顶点染色的对偶问题是一般整数流；在不可定向曲面上, 顶点染色的对偶问题是符号图的整数流.. 本项目主要讨论了符号流与整数流的一些相关问题，研究了符号流的基本性质；讨论了符号模流的存在性以及其与符号整数流之间的关系，给出了符号6-流猜想的等价描述；讨论了符号圆流与符号流之间关系；刻画了没有边不交非平衡圈的符号图，并在这类图上验证了符号6流猜想；刻画了具有小独立数图上流的存在性；刻画了秩为2的向量流；从线性代数角度探讨了一些解决流问题的方法。