Short Directed Cycles in Bipartite Digraphs

Abstract

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 + 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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Acknowledgement

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.

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Correspondence to Sophie Spirkl.

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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

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Mathematics Subject Classification (2010)

  • 05C20
  • 05C35