Some Results On2-factors Of Limit Lengths In Graphs

If there are no special declaration in this paper, the graphs considered in this paper are simple and undirected finite graph.In the graph G, we use V(G) to denote the vertex set of graph G, use E(G) to present the edge set of graph G. In graph G=G(V(G),E(G)), for any v∈V(G), we use d(v, S) to denote the degree of v in subgraph S, use δ(G) to denote the minimum degree of G. P_i denotes a path of order i and length of|V(P)|-1. And C_i denotes a cycle of order i and length of||V(C)|. We use P_=(p1,p2,p3,p4,…,pi)to denote the path Pi, where pt is one vertex of the path. Let Ci=(ci,c2,c3,c4,…, ci) denote the cycle Ci, where ci is one vertex of the cycle. We use|E(P,Q)|to stand for the number of edges between the subgraph P and subgraph Q.A Hamilton cycle of G is the cycle of G which contains every vertex of G. If G′C G and V(G)=V(G’), then we claim G’as the spanning subgraph of G. And the k-factor of G is k-regular spanning subgraph of G, so a2-factor of G is a2-regular spanning subgraph. Easily, every component of the2-factors is a circle. One Hamilton circle of G is a2-factor of G.The path and circle problem in Graph Theory is a very important and active research topic. They have important application in Networks,Biology. Specially, the Hamilton is one of the three famous question in Graph The-ory. Many scholars made much researches on this problem. And the2-factor problem also is one important part in Graph Theory. The2-factor research mainly focuses on the problem the following aspects:the graph of2-factor with specified number, graph of2-factor and path with specified length, graph of2-factor with specified length,etc.We divide this paper into three chapters.In Chapter1,we introduce the development history of Graph Theory.In Chapter2,we introdlice some basic definition in Graph Theory,and introduce the known results of2-factor theory in chronological order.By this way,when read this paper,people can have an overall impression and know the significance of conclusions.The third chapter is the main theorem.In the first section.we mainly proved six lemmas.In the second section,we proved the main theorem.In this part,using the previous lemmas,we prove the result at last.In third section,we propose some problems which can be further discussed,these problems are extended from this paper, and can be used as the next research topic.In this paper,we prove the following result:Theorem3.8Let G be a simple connected graph,|V(G)|=5k.If δ(G)≥3k,then G (?)(k-1)C5∪P5.And the question discussed in this paper comes from the El-Zahar Conjecture.In this conjecture,when n1=n2=…=nk=5,G(?)kG5.In this paper:when n1=n2=…=nk=5,G (?)(k-1)C5∩P5.Conjecture3.8G is a simple graph,|V(G)|=n1+n2+…+nk(ni≥3).If δ(G)≥「n1/2」+「n2/2」+…+「nk/2」,then G has a spanning subgraph consisting of mutually vertex disjoint circlc of lengths n1,n2,…,nk.