Impartial Digraphs

Abstract

We prove a conjecture of Fox, Huang, and Lee that characterizes directed graphs that have constant density in all tournaments: they are disjoint unions of trees that are each constructed in a certain recursive way.

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References

  1. [1]

    O. Antolín Camarena, E. Csóka, T. Hubai, G. Lippner and L. Lovász: Positive graphs, European J. Combin. 52 (2016), 290–301.

    MathSciNet  Article  Google Scholar 

  2. [2]

    D. Conlon, J. Fox and B. Sudakov: An approximate version of Sidorenko’s conjecture, Geom. Funct. Anal. 20 (2010), 1354–1366.

    MathSciNet  Article  Google Scholar 

  3. [3]

    D. Conlon, J. H. Kim, C. Lee and J. Lee: Some advances on Sidorenko’s conjecture, J. Lond. Math. Soc. 98 (2018), 593–608.

    MathSciNet  Article  Google Scholar 

  4. [4]

    D. Conlon and J. Lee: Sidorenko’s conjecture for blow-ups, Discrete Anal., to appear.

  5. [5]

    D. Conlon and J. Lee: Finite reflection groups and graph norms, Adv. Math. 315 (2017), 130–165.

    MathSciNet  Article  Google Scholar 

  6. [6]

    L. N. Coregliano and A. A. Razborov: On the density of transitive tournaments, J. Graph Theory 85 (2017), 12–21.

    MathSciNet  Article  Google Scholar 

  7. [7]

    P. Diaconis and S. Janson: Graph limits and exchangeable random graphs, Rend. Mat. Appl. 28 (2008), 33–61.

    MathSciNet  MATH  Google Scholar 

  8. [8]

    P. Erdős and M. Simonovits: Supersaturated graphs and hypergraphs, Combinatorica 3 (1983), 181–192.

    MathSciNet  Article  Google Scholar 

  9. [9]

    H. Hatami: Graph norms and Sidorenko’s conjecture, Israel J. Math. 175 (2010), 125–150.

    MathSciNet  Article  Google Scholar 

  10. [10]

    J. H. Kim, C. Lee and J. Lee: Two approaches to Sidorenko’s conjecture, Trans. Amer. Math. Soc. 368 (2016), 5057–5074.

    MathSciNet  Article  Google Scholar 

  11. [11]

    J. L. X. Li and B. Szegedy: On the logarithimic calculus and Sidorenko’s conjecture, arXiv:1107.1153.

  12. [12]

    L. Lovász: Large networks and graph limits, American Mathematical Society Colloquium Publications, vol. 60, American Mathematical Society, Providence, RI, 2012.

    Google Scholar 

  13. [13]

    L. Lovász and B. Szegedy; Regularity partitions and the topology of graphons, An irregular mind, Bolyai Soc. Math. Stud., vol. 21, János Bolyai Math. Soc., Budapest, 2010, 415–446.

    Google Scholar 

  14. [14]

    MathOverflow: https://mathoverflow.net/q/189222.

  15. [15]

    A. Sah, M. Sawhney, D. Stoner and Y. Zhao; A reverse Sidorenko inequality, Invent. Math., to appear.

  16. [16]

    A. Sidorenko: A correlation inequality for bipartite graphs, Graphs Combin. 9 (1993), 201–204.

    MathSciNet  Article  Google Scholar 

  17. [17]

    B. Szegedy: An information theoretic approach to Sidorenko’s conjecture, arXiv:1406.6738.

  18. [18]

    E. Thörnblad: Decomposition of tournament limits, European J. Combin. 67 (2018), 96–125.

    MathSciNet  Article  Google Scholar 

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Acknowledgments

We thank the anonymous referees for careful readings and helpful comments.

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Correspondence to Yufei Zhao.

Additional information

Zhao was supported by NSF Awards DMS-1362326 and DMS-1764176, the MIT Solomon Buchsbaum Fund, and a Sloan Research Fellowship. Zhou was supported by the MIT Undergraduate Research Opportunities Program.

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Zhao, Y., Zhou, Y. Impartial Digraphs. Combinatorica 40, 875–896 (2020). https://doi.org/10.1007/s00493-020-4280-0

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Mathematics Subject Classification (2010)

  • 05C20