Cycles in bipartite graphs and an application in number theory

Let G = G(A, B) be a bipartite graph with |A| = u, |B| = v, and let / be a positive integer. In this paper we prove the following result: If v ≤ u, uv ≤ n, m = |E(G)|, and m ≥ max{180/u, 20/n^{1/2(1+(1/l))},} then G contains a C_2/. © 1995 John Wiley & Sons, Inc.