Advertisement

Combinatorica

, Volume 37, Issue 1, pp 49–76 | Cite as

Counting flags in triangle-free digraphs

  • Jan Hladký
  • Daniel Král’
  • Sergey Norin
Original Paper
  • 126 Downloads

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.

Mathematics Subject Classification (2000)

05C35 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    R. Baber and J. Talbot: Hypergraphs do jump, Combin. Probab. Comput. 20 (2011), 161–171.MathSciNetCrossRefMATHGoogle Scholar
  2. [2]
    J. A. Bondy: Counting subgraphs: a new approach to the Caccetta-Häggkvist conjecture, Discrete Math. 165/166 (1997), 71–80.CrossRefMATHGoogle Scholar
  3. [3]
    L. Caccetta and R. Häggkvist: On minimal digraphs with given girth, in: Proceedings of the Ninth Southeastern Conference on Combinatorics, Graph Theory, and Computing (Florida Atlantic Univ., Boca Raton, Fla., 1978), Congress. Numer., XXI, 181-187, Winnipeg, Man., 1978. Utilitas Math.Google Scholar
  4. [4]
    M. Chudnovsky, P. Seymour and B. Sullivan: Cycles in dense digraphs, Combinatorica 28 (2008), 1–18.MathSciNetCrossRefMATHGoogle Scholar
  5. [5]
    J. Cummings, D. Král’, F. Pfender, K. Sperfeld, A. Treglown and M. Young: Monochromatic triangles in three-coloured graphs, J. Combin. Theory Ser. B 103 (2013), 489–503.MathSciNetCrossRefMATHGoogle Scholar
  6. [6]
    S. Das, H. Huang, J. Ma, H. Naves and B. Sudakov: A problem of Erdős on the minimum number of k-cliques, J. Combin. Theory Ser. B 103 (2013), 344–373.MathSciNetCrossRefMATHGoogle Scholar
  7. [7]
    M. Dunkum, P. Hamburger and A. Pór: Destroying cycles in digraphs, Combinatorica 31 (2011), 55–66.MathSciNetCrossRefMATHGoogle Scholar
  8. [8]
    A. Grzesik: On the maximum number of five-cycles in a triangle-free graph, J. Combin. Theory Ser. B 102 (2012), 1061–1066.MathSciNetCrossRefMATHGoogle Scholar
  9. [9]
    P. Hamburger, P. Haxell and A. Kostochka: On directed triangles in digraphs, Electron. J. Combin. 14 Note 19, (electronic), 2007.Google Scholar
  10. [10]
    H. Hatami, J. Hladký, D. Král’, S. Norine and A. Razborov: Non-threecolorable common graphs exist, Combin. Probab. Comput. 21 (2012), 734–742.MathSciNetCrossRefMATHGoogle Scholar
  11. [11]
    H. Hatami, J. Hladký, D. Král’, S. Norine and A. Razborov: On the number of pentagons in triangle-free graphs, J. Combin. Theory Ser. A 120 (2013), 722–732.MathSciNetCrossRefMATHGoogle Scholar
  12. [12]
    H. Hatami and S. Norine: Undecidability of linear inequalities in graph homomorphism densities, J. Amer. Math. Soc. 24 (2011), 547–565.MathSciNetCrossRefMATHGoogle Scholar
  13. [13]
    D. Král’, C.-H. Liu, J.-S. Sereni, P. Whalen and Z. B. Yilma: A new bound for the 2=3 conjecture, Combin. Probab. Comput. 22 (2013), 384–393.MathSciNetCrossRefMATHGoogle Scholar
  14. [14]
    L. Lovász and B. Szegedy: Limits of dense graph sequences, J. Combin. Theory Ser. B 96 (2006), 933–957.MathSciNetCrossRefMATHGoogle Scholar
  15. [15]
    L. Lovász and B. Szegedy: Random graphons and a weak Positivstellensatz for graphs, J. Graph Theory 70 (2012), 214–225.MathSciNetCrossRefMATHGoogle Scholar
  16. [16]
    O. Pikhurko and E. R. Vaughan: Minimum number of k-cliques in graphs with bounded independence number, Combin. Probab. Comput. 22 (2013), 910–934.MathSciNetCrossRefMATHGoogle Scholar
  17. [17]
    A. A. Razborov: Flag algebras, J. Symbolic Logic 72 (2007), 1239–1282.MathSciNetCrossRefMATHGoogle Scholar
  18. [18]
    A. A. Razborov: On the minimal density of triangles in graphs, Combin. Probab. Comput. 17 (2008), 603–618.MathSciNetCrossRefMATHGoogle Scholar
  19. [19]
    A. A. Razborov: On 3-hypergraphs with forbidden 4-vertex configurations, SIAM J. Discrete Math. 24 (2010), 946–963.MathSciNetCrossRefMATHGoogle Scholar
  20. [20]
    A. A. Razborov: On the Caccetta-Häggkvist conjecture with forbidden subgraphs, J. Graph Theory 74 (2013), 236–248.MathSciNetCrossRefMATHGoogle Scholar
  21. [21]
    J. Shen: Directed triangles in digraphs, J. Combin. Theory Ser. B 74 (1998), 405–407.MathSciNetCrossRefMATHGoogle Scholar
  22. [22]
    B. Sullivan: A summary of results and problems related to the Caccetta-Häggkvist conjecture, unpublished, 2006.Google Scholar

Copyright information

© János Bolyai Mathematical Society and Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  1. 1.Institute of Mathematics of the Academy of Sciences of the Czech RepublicPrahaCzech Republic
  2. 2.Mathematics Institute and Department of Computer ScienceUniversity of WarwickCoventryUK
  3. 3.Department of Mathematics & StatisticsMcGill UniversityMontrealCanada

Personalised recommendations