Some Results On Independent Cycles In Graphs

All graphs considered in this paper are finite simple graphs. Let V(G) and E(G) denote the vertex set and edge set of G respectively, then G=(V(G), E(G)) be a graph, We use dG(v) to stand for the degree of every v in G. Letδ(G)=mindG(v)|v∈V(G) denote the minimum vertex degree of G, we also use delta in not cause confusion circumstances. If each vertex degrees are the same, that we call G for the regular graph. For a graph G,|G|=|V(G)|is the order of G, andσ2(G)= min{dG(x)+dG(y)|x,y∈V(G),x≠y,xy(?)E(G)} is the minimum degree sum of nonadjacent vertices of G. (When G is a complete graph, we defineσ2(G)=∞).For a bipartite graph G with partite sets V1 and V2, we defineδ1,1(G)=min{dG(x)+dG(y)|x∈V1,y∈V2},σ1,1(G)=min{dG(x)+dG(y)|x∈V1,y∈V2,xy(?)E(G)}.(When G is a complete bipartite graph, we defineσ1,1=∞).When|V1|=|V2|, we call G=G(V1, V2) a balance bipartite graph. We de-note the length of path P and the length of cycle C by l(P) and l(C), respectively. That is, l(P)=|V(P)|-1,l(C)=|V(C)|. A Hamilton cycle of G is a cycle of G which contains every vertex of G. If has G'C G, causes V(G)=V(G'), then call G'as the spanning subgraph of G, A 1-regular spanning subgraph of Gis a 1-factor of a graph G, and we call a 1-factor a perfect matching. It is obviously that a 1-factor of G is a collection of independent edges that covers all vertices of G. A 2-factor of G is a 2-regular spanning subgraph of G. It is easy to know each component of a 2-factor of G is a cycle. k independent cycles of G arc k cycles which have no common vertex in G. Then the independent cycles,2-factor in G arc important problems with the close relation in graph factorial theory, also they are the extending of Hamilton cycle theory. The discussion is mature and perfect increasingly. And it is important applications to computer science and network.The study of independent cycles,2-factor theory mostly focus on follow-ing:Graph containing specified number of independent cycles and 2-factor, inde-pendent cycles and 2-factor with specified length, graph containing independent cycles and 2-factor which have specified properties, etc.The innovation of this paper is the giving of the proof process. This paper is divided into three chapters. In Chapter 1, we introduce some basic notations and give a concise survey on the history and the progress on the cycle theory, and the primary results about the paper. This chapter is the base of the following three chapters.In Chapter 2, we investigate the graph containing independent cycles which have specified lengths, In this chapter, we have the following result:Theorem2.2.1.若|G|=4k,k≥5,σ2(G)≥4k-3,则G包含k-1个点不相交的4-圈。In Chapter 3, we investigate the graph containing specified length of inde-pendent cycles in bipartite graphs and prove the following result:Theorem3.1.1. G=(V1,V2;E)是二部图,|V1|=|V2|=3k,k是一个正整数,如果σ1,1(G)≥4k-1,那么G包含k-2个6-圈�'�一个12-圈,并且它们�'相独立。...