The Research On 3-flow Conjecture

Graph theory is a jumped-up subject which develops very rapidly in recent years and is applied extensively.Many practical problems can be solved through graphs.The graph coloring problem is one of the most famous NP-complete problems.It is a basic method which can be applied in solving some practical problems about task allocation such as transportation problem,time table prob-lem,circuit design and storage problem.Four colour theorem,one of three famous conjecture,has been solved by computer,however,it isn't accepted by mathemati-cians.By studying on character of the colouring graph,in 1950,Tutte put out that a plan graph is face-k-colorable if and only if it exists a nowhere-zero k-flow.Therefore,he introduced nowhere-zero integer flow theory as a tool to study coloring problem and he proposed some related conjectures.Hence,the coloring problem can be converted into an integer flow problem to research.So the integer flow problem became an important problem in graph theory research.Nowhere-zero 3-flow is a spacial case of nowhere-zero integer flow and a lot of works are related to it.The most familiar one is Tutte' s 3-flow conjecture which said that every 4-edge connected graph admits a nowhere-zero 3-flow.3-flow Con-jecture is regarded as a beautiful conjecture in graph theory world.Many famous mathematicians work on this subject,and get a lot of very nice results.But 3-flow Conjecture has not been solved yet.Kochol proposed another four equiv-alent versions of Tutte's 3-flow conjecture,that is,Every 5-edge-connected graph admits a nowhere-zero 3-flow,and so on.Based on Kochol's research,Chen and N-ing proposed a new equivalent version of Tutte's 3-flow:Every 5-edge-connected graph with minimum degree at least 6 has a nowhere-zero 3-flow.Recently,the Weak 3-flow Conjecture has been solved,which blaze a way in solving 3-flow conjecture.Depending on the known results,we study the strongly Z3-connectivity of graphs,which are stronger than the existence of nowhere-zero 3-flow.At the same time,we investigate the some graphs which satisfy some conditions and admit nowhere-zero 3-flow.In this thesis,the main content is divided into three chapters.In Chapter 1,we firstly introduce some basic concepts and definitions of integer flow,as well as the property of 3-flow and the group connectivity of graphs.In Chapter 2,we give some existing results about 3-flow and the equivalent version of 3-flow,and then we study on the graphs which is 5-edge-connected with no nontrivial 5-edge-cut whether it is strongly Z3-connected in every vertex of degree 5.We extend this result to confirm whether the graph which is 5-edge-connected with minimum degree 6 admits a nowhere-zero 3-flow or not.In chapter 3,we do some job about Zhang's conjecture,we restrict in some conditions and get the result.