均衡二部图中含2k条指定边的k个独立圈及2-因子

得到了对于二部图G=(V_1,V_2;E),当|V_1|=|V_2|=n≥2k+1时的结果:对G中任意2k条独立边e_1,e_1~*,…,e_k,e_k~*,G中一定存在k个独立的4-圈C_1,C_2,…,C_k,使得对任意i∈{1,2,…,k}有{e_i,e_i~*}E(C_i).并在此基础上进一步证明了当|V_1|=|V_2|=n≥3k时若对任意两顶点x∈V_1,y∈V_2,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C_1,C_2,…,C_(k+1)使得对任意i∈{1,2,…,k}有{e_i,e_i~*}E(C_i)且|C_i|=4.

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

This paper obtains the conclution for a bipartite graph G=(V_1,V_2;E)with |V_1|=|V_2|=n such that n≥2k+1 where k≥1 is an integer.That is for any 2k independent edges e_1,e_1~*,…,e_k,e_k~* of G,G contains k vertex-disjoint quadrilaters C_1,C_2,…,C_k such that {e_i,e_i~*}CE(C_i)for each i∈{1,2,…,k}.On the basis of this,We further prove that if|V_1|=|V_2|=n such that n≥3k and d(x)+d(y)≥2n-k+1 for each pair of vertices x and y of G withx∈V_1 and y∈V_2,then G has a 2-factor with k+1vertex-disjoint cycles C_1,C_2,…,C_(k+1)such that {e_i,e_i~*}CE(C_i)for each i∈{1,2,…,k},and |C_i|=4.