Z₃-Connectivity Of 4-Edge Connected Triangle Graph

The integer flow theory was introduced by Tutte as a tool to solve the Four Color Problem.It is well known that a planar graph admits a nowhere-zero k-flow is equivalent to it is face-k-colorable.In the study of integer flow,researchers devoted into the famous three conjectures,5-flow Conjecture,4-flow Conjecture and 3-flow Conjecture.Among these conjectures,3-flow conjecture receives more and more attention.3-flow conjecture was proposed by Tutte in the 1950s,and still remains open today.In 2012,C.Thomassen solved the weak 3-flow conjecture(there is a constant k such that every k-edge-connected graph has a nowhere-zero 3-flow).This significant breakthrough give us hope for solving 3-flow conjecture.During the study of integer flow,F Jaeger proposed another concept,group connectivity,which is a property relative to integer flow and stronger than integer flow.Suppose that a graph is Zk-connected,then it has a nowhere-zero k-flow,but not vice versa.To certify the group connectivity or the existence of integer flow of a graph is believed to be difficult,it is natural for us to focus on some special classes of graphs.Xu and Zhang in[30]proposed a conjecture about the existence of 3-flow of triangular graphs:every 4-edge-connected triangular graphs has a nowhere-zero 3-flow.Hou et al.proved in[9]that every 4-edge-connected 2-triangular graph is Z3-connected(and so it has a nowhere-zero 3-flow).Invoted by this result and the conjecture above,we consider a special case of 4-edge-connected triangular graph,and prove that these graphs are Z3-connected,which improves the result in[9].The main content of this paper is divided into three chapters.In the first chapter,we introduce the concepts of integer flow k-flow and modular k-flow of a graph,and introduce some conclusions and conjectures about integer flow.Among them,we mainly introduce the nowhere-zero integer flow in 2-edge-connected graph and 4-edge-connected graph and triangularly connected graphs.In chapter 2,we introduce some conclusions and conjectures related to integer flow and group connectivity,including some results about cycles,wheels,triangularly connected graphs,square graphs,chordal graphs and some graphs under the degree of edge connectivity conditions.In chapter 3,the presupposition lemmas and corollaries related to the main conclusions of this paper are given.Finally,the main conclusion is proved:for every 4-edge-connected triangle graph G,if each of its triangularly connected component is essentially 4-edge-connected,and every key vertice v satisfies that(?)d(v)?5 and v has degree at least 3 in some triangularly connected components;(?)if v has degree three in some triangulary connected component of G,then N(v)induces a K3 in this triangularly connected component,then G is Z3-connected.