Long cycles in graphs and digraphs

Consider the following problem (solved by Woodall): given , find the minimum size of a graph G guaranteeing the existence of a cycle of length p.We prove that a balanced bipartite graph of order 2n and size greater than n(n−k−1)+k+1 contains a cycle of length 2n−2k, where .In the paper we also formulate and give a solution to an analogous problem for digraphs.