The Caccetta-Häggkvist 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 2 k . 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 α,β > 0 with kα + β > 1, if G is a bipartite digraph with bipartition ( A, B ), where every vertex in A has out-degree at least β |B|, and every vertex in B has out-degree at least α|A |, then G has a directed cycle of length at most 2 k . This implies the Caccetta-Häggkvist conjecture (set β > 0 and very small), and again is best possible for infinitely many pairs ( α,β ). We prove this for k = 1,2, and prove a weaker statement (that α + β > 2/( k + 1) suffices) for k = 3,4.

中文翻译：

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二部有向图中的短有向环

Caccetta-Häggkvist 猜想意味着对于每个整数 k ≥ 1，如果 G 是一个二部有向图，每个部分都有 n 个顶点，并且每个顶点的出度都大于 n /( k +1)，那么 G 有一个有向图循环长度最多为 2 k 。如果为真，这是最好的可能，我们证明了 k = 1, 2, 3, 4, 6 并且所有 k ≥ 224,539。更一般地，我们推测对于每个整数 k ≥ 1 和每对实数 α,β > 0 且 kα + β > 1，如果 G 是一个二分有向图 ( A, B )，其中 A 中的每个顶点都有出度至少为 β |B|，并且 B 中的每个顶点的出度至少为 α|A |，则 G 具有长度最多为 2 k 的有向环。这意味着 Caccetta-Häggkvist 猜想（设置 β > 0 并且非常小），并且对于无限多对 ( α,β ) 也是最好的。我们证明 k = 1,2，