On The Existence Of Vertex-disjoint Chorded Cycles In Bipartite Graph

As a branch of discrete mathematics,graph theory has existed more than 200 years.In recent years,graph theory is extremely active in scientific fields,and it has already shown great advantages to solve problems about computer science,biology,chemistry and other subjects using graph theory.In this paper,we considered simple,finite,undirected graphs with no multiple edges and no loops.In this paper,we mainly consider about tcycles he following problems:The minimum degree condition for the existence of vertex-disjoint chorded containing specified vertices in a bipartite graph,the minimum degree condition for the existence of vertex-disjoint double chorded cycles in a bipartite graph.Let G =(V,E)be a undirected graph,a graph G is bipartite if V can be partitioned into two sets V1 and V2 so that every edge of G joins a vertex of V1 and a vertex of V2.A chord is an edge between two vertices of a cycle that is not an edge on the cycle.If a cycle has at least one chord,then the cycle is called a chorded cycle.If a cycle has at least two chords,then the cycle is called a double chorded cycle.This paper is divided into three chapters.In chapter 1,we introduce some notations and terminology,the history and the progress of the problem of we study.In chapter 2,we discuss the minimum degree condition for the existence of vertex-disjoint chorded cycles containing specified vertices in a bipartite graph.Let G =(Vi,V2;E)be a bipartite graph with |V1| = |V2| = n ? 12k-4,where k is a positive integer.We prove that if ?(G)?n+1/2 then for any k distinct vertices u1,…,uk in G,there exist k disjoint chorded cycles C1,…,Ck in G such that ui?(Ci)and 6?|Ci|? 8 for al 1<i?k.In chapter 3,we discuss the minimum degree condition for the existence of vertex-disjoint double chorded cycles in a bipartite graph.Let G =(Vi,V2;E)be a bipartite graph with |V1| = |V2|?3k,where k is a positive integer.We prove that if ?(G)? 2k + 1,then G admits k disjoint double chorded cycles.