Counting flags in triangle-free digraphs

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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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