Study On 3-flow Problem

Graph theory is a branch of mathematics which is widely used,and graph is its research object.Graph theory has been widely used in many areas of sci-ence and engineering,such as computer science,biological science,VLSI phys-ical design and so on.Graph theory has history of two hundreds years,and it originates from the study of mathematical games,such as Seven Bridges of Konigsberg problem,many popular and difficult game problems.Many mathe-maticians devoted themselves to solving these difficult problems,in succession some mathematical problems have been proposed,such as Four Color Problem.In 1950,Tutte presented that a bridgeless planar graph is face-k-colorable if and only if it exists a nowhere-zero k-flow.As a tool to solve Four Color Problem,integer flow has been introduced and it has been an important project in graph theory.Tutte proposed that every 4-edge connected graph admits a nowhere-zero 3-flow and it is well-known 3-flow conjecture.3-flow conjecture is still open and believed to be difficult,so some researchers focus on some special class-es of graphs.Lai proposed that every 4-edge-connected 2-triangular graph is Z3-connected,therefore every 4-edge-connected 2-triangular graph admits a nowhere-zero 3-flow.However,it has been proved that not all 4-edge-connected triangular graphs are Z3-connected.Hence,Zhang conjectured that every 4-edge-connected triangular graph admits a nowhere-zero 3-flow.In this paper,we confirm the conjecture for 4-edge-connected triangular graphs whose trian-gular connected blocks has no nowhere-zero 3-flow.In this paper,main content is divided into three chapters.In Chapter 1,we introduce many relevant concepts and definitions of k-flow,modular k-flow.Furthermore,we list some known results about integer flow,including 8-flow theorem,6-flow theorem.As extension of the existence of integer flow,group connectivity of graph are introduced in this chapter.In Chapter 2,we list some known results about 3-flow conjecture.Among these results,the solving of weak 3-flow conjecture can be seen as a breakthrough to 3-flow conjecture.Furthermore,we give main conclusion of this paper.In Chapter 3,by presenting and proving some useful lemmas,we finally complete the proof of our main theorem.