Short Directed Cycles in Bipartite Digraphs

The Caccetta-Haggkvist conjecture implies that for every integer k >= 1, if G is a bipartite digraph, with n vertices in each part, and every vertex has out-degree more than n/(k+1), then G has a directed cycle of length at most 2k. If true this is best possible, and we prove this for k = 1, 2, 3, 4, 6 and all k >= 224,539. More generally, we conjecture that for every integer k >= 1, and every pair of reals alpha,beta < 0 with k alpha + beta < 1, if G is a bipartite digraph with bipartition (A, B), where every vertex in A has out-degree at least beta|B|, and every vertex in B has out-degree at least alpha|A|, then G has a directed cycle of length at most 2k. This implies the Caccetta-Haggkvist conjecture (set beta < 0 and very small), and again is best possible for infinitely many pairs (alpha,beta). We prove this for k = 1,2, and prove a weaker statement (that alpha + beta < 2/(k + 1) suffices) for k = 3,4.