均衡二部图中含2k条指定边的k个独立圈及2-因子

2-Factor and k Vertex-Disjoint Quadrilaterals Containing 2k Specified Edges in a Balance Bipartite Graph

得到了对于二部图G=（V1，V2；E），当│V1│=│V2│=n≥2k＋1时的结果：对G中任意2k条独立边e1，e1^＊，…，ek，ek^＊，G中一定存在k个独立的4-圈C1，C2，…，Ck，使得对任意i∈{1，2，…，k）有{ei，ei^＊） E（Gi）．并在此基础上进一步证明了当│V1│=│2│=n≥3k时若对任意两顶点x∈V1，y∈V2，都有d（x）＋d（Y）≥2n—k＋1成立，则G有一个2-因子含有k＋1个独立圈C1，C2，…，Ck＋1使得对任意i∈{1，2，…，k）有{ei，ei^＊} E（Ci）且│Ci│=4．

This paper obtains the conclution for a bipartite graph G = （V1, V2;E） with │V1│ = │V2│= n such that n ≥2k＋1 where k ≥ 1 is an integer. That is for any 2k independent edges e1, e1^＊ ,ek, ek^＊ of G, G contains k vertex-disjoint quadrilaters C1, C2,…, Ck such that {ei, ei^＊} C E（Ci）for each i {1, 2,…. , k}. On the basis of this,We further prove that if │V1│ = │V2│ = n such that n 〉 3k and d（x） ＋ d（y） 〉 2n- k ＋ 1 for each pair of vertices x and y of G withx ∈ V1 and y ∈ V2, then G has a 2-factor with k ＋ lvertex-disjoint cycles C1, C2,… , Ck＋1 such that {ei, ei^＊} E （Ci） for each i {1, 2,… , k}, and │Ci│ = 4.