二分图中含有完美对集的2因子

该文证明若G是2n阶均衡二分图，δ(G)≥(2n-1)/3，则对任何正整数k，n≥4k时，任给G的一个完美对集M，G中存在一个包含M的所有边的恰含k个分支的2 因子(k=1，n=5且δ(G)=3除外）. 特别k=2时，在条件n≥5且δ(G)≥(n+2)/2下，结论也成立. 这里所给的δ(G)的下界是最好的可能.

On 2 factors Containing Perfect Matching in Bipartite Graphs

A bipartite graph G=(X,Y; E) is called  balanced if |X|=|Y|. Let  G=(X,Y; E) be a balanced bipartite  graph of order 2n, suppose  that the minimum degree of G is at least   (2n-1)/3, the author shows that if n≥4k, then for each perfect  matching M, G contains a 2 factor with exactly k components  (vertex disjoint cycles) including every edge of M (with one  exception that k=1, n=5 and δ(G)=3). When k=2,  n≥5,  the author has  the same conclusionunder  the condition  4δ≥(n+1)/2,  and  this bound about minimum degree  is the best possible.