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An example of Tateno disproving conjectures of Bonato–Tardif, Thomasse, and Tyomkyn

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Abstract

In his 2008 thesis [16] , Tateno claimed a counterexample to the Bonato–Tardif conjecture regarding the number of equimorphy classes of trees. In this paper we revisit Tateno’s unpublished ideas to provide a rigorous exposition, constructing locally finite trees having an arbitrary finite number of equimorphy classes; an adaptation provides partial orders with a similar conclusion. At the same time these examples also disprove conjectures by Thomassé and Tyomkyn.

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References

  1. Abdi, D.: Siblings of binary relations: the case of direct sums of chains, NE-free posets and trees. Ph.D. Thesis, University of Calgary (2022)

  2. Bonato, A., Tardif, C.: Mutually embeddable graphs and the tree alternative conjecture. J. Combin. Theory Ser. B 96(6), 874–880 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  3. Bonato, A., Bruhn, H., Diestel, R., Sprüssel, P.: Twins of rayless graphs. J. Combin. Theory Ser. B 101(1), 60–65 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  4. Braunfeld, S., Laskowski, C.: Counting siblings in universal theories. J. Symb. Log. 87 (2022), no. 3, 1130–1155

  5. Diestel, R.: Graph Theory. Graduate Texts in Mathematics, 4th edn., vol. 173, p. xviii+437. Springer, Heidelberg (2010)

  6. Hahn, G., Pouzet, M., Woodrow, R.: Siblings of countable cographs. arXiv preprint arXiv:2004.12457v1 (2020)

  7. Halin, R.: Fixed configurations in graphs with small number of disjoint rays. In: Bodendiek, R. (ed.) Contemporary Methods in Graph Theory, pp. 639–649. Bibliographisches Inst, Mannheim (1990)

    Google Scholar 

  8. Halin, R.: Automorphisms and endomorphisms of infinite locally finite graphs. Abh. Math. Sem. Univer. Hamburg 39, 251–283 (1973)

    Article  MathSciNet  MATH  Google Scholar 

  9. Halin, R.: The structure of rayless graphs. Abh. Math. Sem. Univer. Hamburg 68, 225–253 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  10. Hamann, M.: Self-embeddings of trees. Discrete Math. 342(12), 111586 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  11. Imrich, W.: Subgroup theorems and graphs. In: Combinatorial Mathematics, V (Proc. 5th Austral. Conf., Roy. Melbourne Inst. Tech., Melbourne, 1976). Lecture Notes in Math., vol. 622, pp. 1–27. Springer, Berlin (1977)

  12. Laflamme, C., Pouzet, M., Woodrow, R.: Equimorphy—the case of chains, archive for mathematical logic. Arch. Math. Logic 56(7–8), 811–829 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  13. Laflamme, C., Pouzet, M., Sauer, N.: Invariant subsets of scattered trees and the tree alternative property of Bonato and Tardif. Abh. Math. Semin. Univer. Hambg. 87(2), 369–408 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  14. Laflamme, C., Pouzet, M., Sauer, N., Woodrow, R.: Siblings of an \(\mathfrak{N}_0\)–categorical relational structure. Contrib. Discrete Math.16(2021), no.2, 90–127

  15. Polat, N., Sabidussi, G.: Fixed elements of infinite trees. Graphs and combinatorics (Lyon, 1987; Montreal, PQ, 1988). Discrete Math. 130(1–3), 97–102 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  16. Tateno, A.: Problems in finite and infinite combinatorics. Ph.D Thesis, Oxford University (2008)

  17. Thomassé, S.: Conjectures on countable relations, circulating manuscript, 17p. 2000, and personal communication (2012)

  18. Tits, J.: Sur le groupe des automorphismes d’un arbre. In: Essays on topology and related topics, pp. 188–211. Springer, New-York (1970)

  19. Tyomkyn, M.: A proof of the rooted tree alternative conjecture. Discrete Math. 309, 5963–5967 (2009)

    Article  MathSciNet  MATH  Google Scholar 

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Author notes

  1. Claude Laflamme and Robert Woodrow are supported by NSERC of Canada.

Authors and Affiliations

  1. Department of Mathematics and Statistics, University of Calgary, Calgary, AB, T2N1N4, Canada

    Davoud Abdi Kalow, Claude Laflamme, Atsushi Tateno & Robert Woodrow

  2. London, United Kingdom

    Atsushi Tateno

Authors
  1. Davoud Abdi Kalow
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  2. Claude Laflamme
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  3. Atsushi Tateno
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  4. Robert Woodrow
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Corresponding author

Correspondence to Claude Laflamme.

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Communicated by Nathan Bowler.

Dedicated to our parents.

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Kalow, D.A., Laflamme, C., Tateno, A. et al. An example of Tateno disproving conjectures of Bonato–Tardif, Thomasse, and Tyomkyn. Abh. Math. Semin. Univ. Hambg. 93, 99–131 (2023). https://doi.org/10.1007/s12188-023-00270-0

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  • DOI: https://doi.org/10.1007/s12188-023-00270-0

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