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Oriented Cycles in Digraphs of Large Outdegree

Combinatorica volume 42, pages 1145–1187 (2022)Cite this article

Abstract

In 1985, Mader conjectured that for every acyclic digraph F there exists K = K(F) such that every digraph D with minimum out-degree at least K contains a subdivision of F. This conjecture remains widely open, even for digraphs F on five vertices. Recently, Aboulker, Cohen, Havet, Lochet, Moura and Thomassé studied special cases of Mader’s problem and made the following conjecture: for every ℓ ≥ 2 there exists K = K(ℓ) such that every digraph D with minimum out-degree at least K contains a subdivision of every orientation of a cycle of length ℓ.

We prove this conjecture and answer further open questions raised by Aboulker et al.

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Acknowledgement

The research on this project was initiated during a joint research workshop of Tel Aviv University and the Freie Universität Berlin on Ramsey Theory, held in Tel Aviv in March 2020, and partially supported by GIF grant G-1347-304.6/2016. We would like to thank the German-Israeli Foundation (GIF) and both institutions for their support.

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Authors and Affiliations

  1. Department of Mathematics, ETH, Zürich, Switzerland

    Lior Gishboliner

  2. Institute of Mathematics, Technische Universität Berlin, Berlin, Germany

    Raphael Steiner

  3. Departement of Computer Science, ETH, Zürich, Switzerland

    Raphael Steiner

  4. Institute of Mathematics, Freie Universität Berlin, Berlin, Germany

    Tibor Szabó

Authors
  1. Lior Gishboliner
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  2. Raphael Steiner
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  3. Tibor Szabó
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Corresponding author

Correspondence to Raphael Steiner.

Additional information

During the work on this project, the author was supported by ERC Starting Grant 633509.

Funded by DFG-GRK 2434 Facets of Complexity.

Research supported in part by GIF grant No. G-1347-304.6/2016 and by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany’s Excellence Strategy - The Berlin Mathematics Research Center MATH+ (EXC-2046/1, project ID: 390685689).

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Gishboliner, L., Steiner, R. & Szabó, T. Oriented Cycles in Digraphs of Large Outdegree. Combinatorica 42 (Suppl 1), 1145–1187 (2022). https://doi.org/10.1007/s00493-021-4750-z

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  • DOI: https://doi.org/10.1007/s00493-021-4750-z

Mathematics Subject Classification (2010)

  • 05C07
  • 05C20
  • 05C83