In this paper, we obtain the following result: Let k, n 1 and n 2 be three positive integers, and let G = (V 1,V 2;E) be a bipartite graph with |V1| = n 1 and |V 2| = n 2 such that n 1 ? 2k + 1, n 2 ? 2k + 1 and |n 1 ? n 2| ? 1. If d(x) + d(y) ? 2k + 2 for every x ∈ V 1 and y ∈ V 2 with xy $ \notin $ E(G), then G contains k independent cycles. This result is a response to Enomoto’s problems on independent cycles in a bipartite graph.
Plesnik in 1972 proved that an （m - 1）-edge connected m-regular graph of even order has a 1-factor containing any given edge and has another 1-factor excluding any given m - 1 edges. Alder et al. in 1999 showed that if G is a regular （2n ＋ 1）-edge-connected bipartite graph, then G has a 1-factor containing any given edge and excluding any given matching of size n. In this paper we obtain some sufficient conditions related to the edge-connectivity for an n-regular graph to have a k-factor containing a set of edges and （or） excluding a set of edges, where 1 ≤ k ≤n/2. In particular, we generalize Plesnik＇s result and the results obtained by Liu et al. in 1998, and improve Katerinis＇ result obtained 1993. Furthermore, we show that the results in this paper are the best possible.
