Counting flags in triangle-free digraphs

Combinatorica volume 37, pages49–76(2017)Cite this article

Abstract

Motivated by the Caccetta-Häggkvist Conjecture, we prove that every digraph on n vertices with minimum outdegree 0:3465n contains an oriented triangle. This improves the bound of 0:3532n of Hamburger, Haxell and Kostochka. The main new tool we use in our proof is the theory of flag algebras developed recently by Razborov.

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Author information

Affiliations

  1. Institute of Mathematics of the Academy of Sciences of the Czech Republic, Žitná 25, Praha, Czech Republic

    Jan Hladký

  2. Mathematics Institute and Department of Computer Science, University of Warwick, Coventry, CV4 7AL, UK

    Daniel Král’

  3. Department of Mathematics & Statistics, McGill University, Montreal, QC, H3A 2K6, Canada

    Sergey Norin

Authors
  1. Jan Hladký
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  2. Daniel Král’
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  3. Sergey Norin
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Corresponding author

Correspondence to Jan Hladký.

Additional information

Institute of Mathematics of the Academy of Sciences of the Czech Republic is supported by RVO:67985840.

JH and DK were partially supported by Grant Agency of Charles University, grant GAUK 202-10/258009. SN was supported in part by NSF under Grant No. DMS-0701033 and an NSERC discovery grant. An extended abstract containing this result appeared in the proceedings of the EuroComb 2009 conference. A major revision of the paper was done during a visit of the first two authors to the Institut Mittag-Leffer (Djursholm, Sweden).

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Hladký, J., Král’, D. & Norin, S. Counting flags in triangle-free digraphs. Combinatorica 37, 49–76 (2017). https://doi.org/10.1007/s00493-015-2662-5

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Mathematics Subject Classification (2000)

  • 05C35