dc.contributor.author | Seymour, Paul | |
dc.contributor.author | Spirkl, Sophie | |
dc.date.accessioned | 2022-08-12 00:35:16 (GMT) | |
dc.date.available | 2022-08-12 00:35:16 (GMT) | |
dc.date.issued | 2020-08-01 | |
dc.identifier.uri | https://doi.org/10.1007/s00493-019-4065-5 | |
dc.identifier.uri | http://hdl.handle.net/10012/18517 | |
dc.description | This is a post-peer-review, pre-copyedit version of an article published in Combinatorica. The final authenticated version is available online at: https://doi.org/10.1007/s00493-019-4065-5 | en |
dc.description.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 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. | en |
dc.description.sponsorship | Supported by ONR grant N00014-14-1-0084 and NSF grant DMS-1265563 | en |
dc.language.iso | en | en |
dc.publisher | Springer Nature | en |
dc.subject | Caccetta–Haggkvist conjecture | en |
dc.subject | bipartite digraph | en |
dc.title | Short Directed Cycles in Bipartite Digraphs | en |
dc.type | Article | en |
dcterms.bibliographicCitation | Seymour, P., Spirkl, S. Short Directed Cycles in Bipartite Digraphs. Combinatorica 40, 575–599 (2020). https://doi.org/10.1007/s00493-019-4065-5 | en |
uws.contributor.affiliation1 | Faculty of Mathematics | en |
uws.contributor.affiliation2 | Combinatorics and Optimization | en |
uws.typeOfResource | Text | en |
uws.peerReviewStatus | Reviewed | en |
uws.scholarLevel | Faculty | en |