图的结构与处处非零3-流及Z3-连通性研究

This project mainly studies two kinds of typical problems in graph theory: the problems on integer flows due to four-color conjecture and the related problems on group connectivity. Around the well-known 3-flow conjecture and the Z3-connectivity conjecture, we shall study nowhere-zero 3-flows and Z3-connectivity in graphs with some special structure by contraction, vertex-splitting, induction and contradiction methods and want to solve the following three questions: (1) Nonwhere-zero 3–flows of graphs satisfying a slightly stronger Chvatal-Erdos condition; (2) Z3-connectivity of 4-edge-connected triangular graphs in which each edge contained in at most two triangles; .(3) Nonwhere-zero 3–flows and Z3-connectivity of 4-edge-connected claw-free graphs without some forbidden subgraph. The study of these questions not only verified the well-known 3-flow conjecture holds for graphs with some special structure, but also laid a theoretical foundation for follow-up studies.

本项目拟主要研究图论中的两类典型问题：源于四色猜想的整数流问题以及与之相关的群连通性问题。围绕著名的3-流猜想和Z3-连通性猜想，我们拟采用收缩、点分裂、归纳、反证等方法，研究具有某种特殊结构的图的处处非零3-流及Z3-连通性，并希望解决以下三个问题:（1）满足较强的 Chvatal-Erdos 条件的图的处处非零3-流问题；（2）每条边包含在至多2个三角形中的4-边连通triangular图的Z3-连通性；（3）不含某种禁用子图的4-边连通无爪图的处处非零3-流及Z3-连通性。这些问题的研究不仅验证了3-流猜想对于具有某种特殊结构的图是成立的，而且为后续的研究奠定了理论基础。

本项目主要研究了图论中的两类典型问题：源于四色猜想的整数流问题以及与之相关的群连通性问题。围绕著名的3-流猜想和Z3-连通性猜想，我们采用收缩、点分裂、归纳、反证等方法，研究具有某种特殊结构的图的处处非零3-流及Z3-连通性，主要解决了以下三个问题:（1）满足较强的Chvatal-Erdos条件且独立点数为3 的图的处处非零3-流及Z3-连通性；（2）每条边包含在至多2个三角形中的4-边连通triangular-图的Z3-连通性；（3）不含某种禁用子图的4-边连通无爪图的处处非零3-流及Z3-连通性。这些问题的研究不仅验证了3-流猜想对于具有某种特殊结构的图是成立的，而且为后续的研究奠定了理论基础。