A minimum degree result for disjoint cycles and forests in bipartite graphs

Let F = (U1, U2; W ) be a forest with |U1| = |U2| = s, where s ≥ 2, and let G = (V1, V2, E) be a bipartite graph with |V1| = |V2| = n ≥ 2k + s, where k is a nonnegative integer. Suppose that the minimum degree of G is at least k + s. We show that if n > 2k + s then G contains the disjoint union of the forest F and k disjoint cycles. Moreover, if n = 2k + s, then G contains the disjoint union of the forest F , k − 1 disjoint cycles and a path of order 4.