Abstract
A tree is scattered if it does not contain a subdivision of the complete binary tree as a subtree. We show that every scattered tree contains a vertex, an edge, or a set of at most two ends preserved by every embedding of T. This extends results of Halin, Polat and Sabidussi. Calling two trees equimorphic if each embeds in the other, we then prove that either every tree that is equimorphic to a scattered tree T is isomorphic to T, or there are infinitely many pairwise non-isomorphic trees which are equimorphic to T. This proves the tree alternative conjecture of Bonato and Tardif for scattered trees, and a conjecture of Tyomkyn for locally finite scattered trees.
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References
Bonato, A., Tardif, C.: Mutually embeddable graphs and the tree alternative conjecture. J. Combin. Theory Ser. B 96(6), 874–880 (2006)
Bonato, A., Bruhn, H., Diestel, R., Sprüssel, P.: Twins of rayless graphs. J. Combin. Theory Ser. B 101(1), 60–65 (2011)
R. Diestel, Graph theory. Fourth edition. Graduate Texts in Mathematics, 173. Springer, Heidelberg, 2010. xviii+437 pp
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)
Halin, R.: Automorphisms and endomorphisms of infinite locally finite graphs. Abh. Math. Sem. Univ. Hamburg 39, 251–283 (1973)
Halin, R.: The structure of rayless graphs. Abh. Math. Sem. Univ. Hamburg 68, 225–253 (1998)
Hamann, M.: Personnal communication (2016)
Hamann, M.: Group action on metric spaces: fixed points and free subgroups, this volume
Hamann, M.: Self-embeddings of trees. Manuscript (2016)
Jung, H.A.: Wurzelbäume und undendliche Wege in Graphen. Math. Nachr. 41, 1–22 (1969)
König, D.: Theorie der Endlichen und Unendlichen Graphen: Kombinatorische Topologie der Streckenkomplexe. Akad. Verlag, Leipzig (1936)
Laflamme, C., Pouzet, M., Sauer, N., Zaguia, I.: Pairs of orthogonal countable ordinals. Discrete Maths 335, 35–44 (2014)
Laver, R.: Better-quasi-orderings and a class of trees. Studies in foundations and combinatorics, pp. 31–48, Adv. in Math. Suppl. Stud., 1, Academic Press, New York-London (1978)
Lehner, F.: Personnal communication (2016)
Pays, I., Valette, A.: Sous-groupes libres dans les groupes d’automorphismes d’arbres. L’enseignement Mathématique 37, 151–174 (1991)
Polat, N., Sabidussi, G.: Fixed elements of infinite trees, in Graphs and combinatorics (Lyon, 1987; Montreal, PQ, 1988). Discrete Math. 130(1–3), 97–102 (1994)
Tits, J.: Sur le groupe des automorphismes d’un arbre, Essays on topology and related topics, pp 188–211, Springer, New-York (1970)
Tyomkyn, M.: A proof of the rooted tree alternative conjecture. Discrete Math. 309, 5963–5967 (2009)
Woess, W.: Fixed sets and free subgroups of groups acting on metric spaces. Math. Z. 214, 425–440 (1993)
Acknowledgements
C. Laflamme was supported by NSERC of Canada Grant # 10007490. Research started while M. Pouzet visited the Mathematics and Statistics Department of the University of Calgary in June 2012; the support provided is gratefully acknowledged. N. Sauer was supported by NSERC of Canada Grant # 10007490
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C. Laflamme was supported by NSERC of Canada Grant # 10007490. Research started while M. Pouzet visited the Mathematics and Statistics Department of the University of Calgary in June 2012; the support provided is gratefully acknowledged. N. Sauer was supported by NSERC of Canada Grant # 10007490.
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Laflamme, C., Pouzet, M. & Sauer, N. Invariant subsets of scattered trees and the tree alternative property of Bonato and Tardif. Abh. Math. Semin. Univ. Hambg. 87, 369–408 (2017). https://doi.org/10.1007/s12188-016-0169-7
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DOI: https://doi.org/10.1007/s12188-016-0169-7