Graphs and Combinatorics (1997) 13: 267-273 Graphs and Combinatorics © Springer-Verlag 1997 On a Conjecture on Directed Cycles in a Directed Bipartite Graph Charles Little l , Kee Teo l , and Hong Wang 2 I Department of Mathematics, Massey University, Palmerston North, New Zealand 2 Department of Mathematics, University of New Orleans, New Orleans, LA 70148, USA e-mail: hwang @math.uno.edu Abstract. Let D = (Vt, V 2 ; A) be a directed bipartite graph with I Vtl = I V 2 1 = n 2. Suppose that dD(x) + dD(y) 3n for all x E VI and y E V 2 • Then, with one exception, D contains two vertex-disjoint directed cycles of lengths 2n l and 2n2' respectively, for any positive integer partition n = n t + n2. This proves a conjecture proposed in [9]. 1. Introduction We discuss only finite simple graphs and strict directed graphs. The terminology and notation concerning graphs is that of [4J, except as indicated. A directed graph D is called a directed bipartite graph ifthere exists a partition {VI' V 2 } of V(D) such that the two induced directed subgraphs D[VIJ and D[V 2J of D contain no arcs of D. We denote by (VI' V 2 ;A) a directed bipartite graph with {VI' V 2 } as its biparti- tion and A as its arc set. Similarly, (VI' V 2;E) represents a bipartite graph with {VI' V 2 } as its bipartition and E as its edge set. The main theorem of this paper proves a conjecture proposed in [9]. This conjecture is as follows: Conjecture [9]. Let D = (VI' V 2 ; A) be a directed bipartite graph with IVII = 1V21 = n 2. Suppose that n is odd and dD(x) + dD(y) 3n for all x E VI and y E V 2 . Then D contains two vertex-disjoint directed cycles of lengths 2n l and 2n 2, respectively, for any positive integer partition n = n l + n2. In [9J, it is proved that when dD(x) + dD(y) 3n + 1 for all x E VI and y E V 2 , the conclusion of the conjecture is true for any integer n 2. This result is sharp- ened in this paper. To state our result, we construct a directed bipartite graph B; of order 2n for every integer n 2 as follows. We use K:. b to denote the complete directed bipartite graph (VI' V 2 ; A) with IVII = a and 1V21 = b such that both (x,y) and (y,x) belong to A for all x E VI and y E V 2 . Let D I = (XI' YI;Ad and D 2 = (X 2, Y 2 ;A 2) be two vertex-disjoint directed bipartite graphs such that D I is isomorphic to and D 2 is isomorphic to Then B; consists of D I and D 2 and all arcs (u, v) and (x,y) for u E Xl' V E Y 2, X E Y I and y E X 2 . We prove the following .