Edge-connectivity Of Undirected Graphs And Disjoint Cycles Of Digraphs

In graph theory,connectivity has always been the focus of graph theorists.The connectivity of graph is an important parameter to measure its stability.In digraphs.the disjoint cycle problem of classical graphs such as regular graphs and tournament graphs is a hot research topics in recent years.In this paper,we mainly focus on these problems,which are divided into three chapters.The specific contents are as followsIn chapter 1,we introduce definitions,symbols and related theorem lemmas that are used in this paper,and summarize the research progress on edge-connectivity and disjoint cycle problemsIn chapter 2,a connected graph G is maximally edge-connectivity if λ=δ.An edge-cut of a connected graph G is a set of edges whose removal disconnects G.G is called super edge-connectivity if every minimum edge-cut consists of edges incident with a vertex of minimum degree.The clique number ω(G)of a graph G is the maximum cardinality of a complete subgraph of G.In this chapter,the sufficient conditions for a 4-clique-free graph to be maximaliy edge-connectivity and super edge-connectivity are given respectively according to the inverse index(R(G)=∑u∈V(G)1/(dG(u))),which generalize the results of dankelmann et alIn chapter 3,we consider a simple digraph D,where the girth is the length of a shortest cycle,denoted as g.A digraph D is called k-regular if dD+(v)=dD-(v)=k for every vertex v ∈ V.Ngo Dac Tan proposed a conjecture in 2017:for every integer g≥ 3,there are only finitely many 3-regular digraphs of girth g without two disjoint cycles of different lengths.In this paper,we mainly prove the existence of disjoint cycles of different lengths in 3-regular digraphs with girth 5,which generalize the part of results of Ngo Dac Tan.