Long cycles in graphs and digraphs

Consider the following problem (solved by Woodall): given p > | G | 2 , find the minimum size of a graph G guaranteeing the existence of a cycle of length p. ve that a balanced bipartite graph of order 2n and size greater than n ( n − k − 1 ) + k + 1 contains a cycle of length 2 n − 2 k , where n ⩾ 1 2 k 2 + 3 2 k + 4 . paper we also formulate and give a solution to an analogous problem for digraphs.