Degree Conditions For The Decomposition Of Vertex-disjoint Cycles In Bipartite Graphs

Graph theory,as a branch of combinatorial mathematics,has a long history.The research object of this paper is limited to bipartite graphs.In particular,if the number of edges between any two vertices in a multiple bipartite graph G is at most 2,then G is said to be a standard multiple bipartite graph.We called the set of cycles in a bipartite graph is independent if and only if no two of them have any common vertex.And the problem of the independent cycles(the disjoint cycles)is that the graph and the existence of chorded cycles is one of the important problems in graph theory.Thus,in this paper,we mainly studied the existence of independent 4 long weight cycles in the standard multiple bipartite graph and the existence of independent chorded cycles in the bipartite graph,and gave the corresponding sufficient conditions.This paper is divided into three chapters.The first chapter mainly introduced the simple terms and explained the background of the research problem.In the second chapter,we mainly studied the problem of independent weight 4 cycles in the standard multiple bipartite graphs:let M=(Vi,V2;E)be a standard multiple bipartite graph with |V1|=?V2??n>2,where n is a positive integer,We prove that if the minimum degree of M is at least 32,then M cotains(?)independent quadrilatearls.Moreover,if n is odd,then(?)of(?)of(?)independent quadrilaterals has four multiedges and the rest one of them has at least three multiedges;if n is even,then(?)of(?)independent quadrilaterals has four multiedges and each of the rest of two has at least three multiedges,with only one exception.And we also raises a farther problem to be discussed.The third chapter mainly proves:Let G=(Vi,V2;E)be a bipartite graph with 3k?m1=?V1???V2?=n,where m,n,k be a positive integer.We prove that if the number of edges of G is at least(4k-2)(n-2)+ 2m,then G contains k independent chorded cycles.