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A Note on the Inducibility of \(4\)-Vertex Graphs

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Abstract

There is much recent interest in understanding the density at which constant size graphs can appear in a very large graph. Specifically, the inducibility of a graph \(H\) is its extremal density, as an induced subgraph of \(G\), where \(|G| \rightarrow \infty \). Already for \(4\)-vertex graphs many questions are still open. Thus, the inducibility of the \(4\)-path was addressed in a construction of Exoo (Ars Combin 22:5–10, 1986), but remains unknown. Refuting a conjecture of Erdős, Thomason (Combinatorica 17(1):125–134, 1997) constructed graphs with a small density of both \(4\)-cliques and \(4\)-anticliques. In this note, we merge these two approaches and construct better graphs for both problems.

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References

  1. Bollobás, B., Egawa, Y., Harris, A., Jin, G.: The maximal number of induced \(r\)-partite subgraphs. Graphs Combin. 11(1), 1–19 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  2. Bollobás, B., Nara, C., Tachibana, S.: The maximal number of induced complete bipartite graphs. Discrete Math. 62(3), 271–275 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  3. Brown, J., Sidorenko, A.: The inducibility of complete bipartite graphs. J. Graph Theory 18(6), 629–645 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  4. Bubeck, S., Linial, N.: On the local profiles of trees. arXiv:1310.5396 (2013, arXiv preprint)

  5. Conlon, D.: On the Ramsey multiplicity of complete graphs. Combinatorica 32(2), 171–186 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  6. Deza, A., Franek, F., Liu, M.: On a conjecture of Erdős for multiplicities of cliques. J. Discrete Algorithms 17, 9–14 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  7. Erdős, P.: On the number of complete subgraphs contained in certain graphs. Magyar Tud. Akad. Mat. Kutató Int. Közl 7, 459–464 (1962)

  8. Erdős, P., Lovász, L., Spencer, J.: Strong independence of graphcopy functions. In: Graph Theory and Related Topics, pp. 165–172 (1979)

  9. Evans, R., Pulham, J., Sheehan, J.: On the number of complete subgraphs contained in certain graphs. J. Combin. Theory Ser. B 30(3), 364–371 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  10. Exoo, G.: Dense packings of induced subgraphs. Ars Combin. 22, 5–10 (1986)

    MathSciNet  MATH  Google Scholar 

  11. Falgas-Ravry, V., Vaughan, E.: Applications of the semi-definite method to the Turán density problem for 3-graphs. Combin. Probab. Comput. 22(01), 21–54 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  12. Franek, F.: On Erdős’s conjecture on multiplicities of complete subgraphs - lower upper bound for cliques of size 6. Combinatorica 22(3), 451–454 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  13. Franek, F., Rödl, V.: 2-colorings of complete graphs with a small number of monochromatic \(K_4\) subgraphs. Discrete Math. 114(1), 199–203 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  14. Giraud, G.: Sur le probleme de Goodman pour les quadrangles et la majoration des nombres de Ramsey. J. Combin. Theory Ser. B 27(3), 237–253 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  15. Goldreich, O., Goldwasser, S., Ron, D.: Property testing and its connection to learning and approximation. J. ACM 45(4), 653–750 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  16. Goodman, A.: On sets of acquaintances and strangers at any party. American Mathematical Monthly, pp. 778–783 (1959)

  17. Hatami, H., Hirst, J., Norine, S.: The inducibility of blow-up graphs. arXiv:1108.5699 (2011, arXiv preprint)

  18. Hirst, J.: The inducibility of graphs on four vertices. J. Graph Theory 75(3), 231–243 (2013)

    Article  MathSciNet  Google Scholar 

  19. Huang, H., Linial, N., Naves, H., Peled, Y., Sudakov, B.: On the densities of cliques and independent sets in graphs. arXiv:1211.4532 (2012, arXiv preprint)

  20. Huang, H., Linial, N., Naves, H., Peled, Y., Sudakov, B.: On the 3-local profiles of graphs. J. Graph Theory 76(3), 236–248 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  21. Jagger, C., Šťovíček, P., Thomason, A.: Multiplicities of subgraphs. Combinatorica 16(1), 123–141 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  22. Linial, N., Morgenstern, A.: Graphs with few 3-cliques and 3-anticliques are 3-universal. J. Graph Theory (2014). doi:10.1002/jgt.21801

  23. Linial, N., Morgenstern, A.: On the number of 4-cycles in a tournament. arXiv:1405.1576 (2014, arXiv preprint)

  24. Lovász, L.: Large Networks and Graph Limits. American Mathematical Society, Providence (2012)

  25. Nieß, S.: Counting monochromatic copies of \(K_4\): a new lower bound for the Ramsey multiplicity problem. arXiv:1207.4714 (2012, arXiv preprint)

  26. Pippenger, N., Golumbic, M.: The inducibility of graphs. J. Combin. Theory Ser. B 19(3), 189–203 (1975)

    Article  MathSciNet  MATH  Google Scholar 

  27. Razborov, A.: Flag algebras. J. Symbol. Logic 72(4), 1239–1282 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  28. Razborov, A.: Flag algebras: an interim report. Math. Paul Erdős II, 207–232 (2013)

  29. Shen, S.: On the parallelization of a search for counterexamples to a conjecture of Erdős. McMaster University Library (2012)

  30. Širáň, J.: A new lower bound for the inducibility of a graph. Math. Slovaca 34(4), 365–370 (1984)

    MathSciNet  MATH  Google Scholar 

  31. Sperfeld, K.: On the minimal monochromatic \(K_4\)-density. arXiv:1106.1030 (2011, arXiv preprint)

  32. Thomason, A.: A disproof of a conjecture of Erdős in ramsey theory. J. Lond. Math. Soc. 2(2), 246–255 (1989)

    Article  MathSciNet  Google Scholar 

  33. Thomason, A.: Graph products and monochromatic multiplicities. Combinatorica 17(1), 125–134 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  34. Vaughan, E.: Flagmatic: a tool for researchers in extremal graph theory (version 2.0) (2013). http://flagmatic.org/graph.html

  35. Wilf, H.S.: The patterns of permutations. Discrete Math. 257(2), 575–583 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  36. Wolf, J.: The minimum number of monochromatic 4-term progressions in \(\mathbb{Z}_p\). J. Combin. 1(1), 53–68 (2010)

    Article  MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, Hebrew University, 91904 , Jerusalem, Israel

    Chaim Even-Zohar

  2. Department of Computer Science, Hebrew University, 91904 , Jerusalem, Israel

    Nati Linial

Authors
  1. Chaim Even-Zohar
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  2. Nati Linial
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Correspondence to Chaim Even-Zohar.

Additional information

Supported by grants from the ERC and from the ISF.

Appendix: The Inducibility of 5-Vertex Graphs

Appendix: The Inducibility of 5-Vertex Graphs

Table 2 Inducibilities of graphs on 5 vertices
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Even-Zohar, C., Linial, N. A Note on the Inducibility of \(4\)-Vertex Graphs. Graphs and Combinatorics 31, 1367–1380 (2015). https://doi.org/10.1007/s00373-014-1475-4

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