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 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 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 2k. 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.
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We would like to thank Alex Scott for his help in classifying the good and bad points of the plane mentioned in section 2, and Farid Bouya for pointing out a mistake in an earlier version of this paper. Our thanks go also to the anonymous referees for their helpful suggestions, which included a simplified proof for 5.3.
Supported by ONR grant N00014-14-1-0084 and NSF grant DMS-1265563.
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Seymour, P., Spirkl, S. Short Directed Cycles in Bipartite Digraphs. Combinatorica 40, 575–599 (2020). https://doi.org/10.1007/s00493-019-4065-5
Mathematics Subject Classification (2010)