Abstract
This note presents a new, elementary proof of a generalization of a theorem of Halin to graphs with unbounded degrees, which is then applied to show that every connected, countably infinite graph G, with \(\aleph _0 \le |{\text {Aut}}(G)| < 2^{\aleph _0}\) and subdegree-finite automorphism group, has a finite set F of vertices that is setwise stabilized only by the identity automorphism. A bound on the size of such sets, which are called distinguishing, is also provided. To put this theorem of Halin and its generalization into perspective, we also discuss several related non-elementary, independent results and their methods of proof.
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Notes
A permutation group is subdegree-finite if its point stabilizers have finite orbits.
Such sets are called distinguishing sets.
References
Albertson, M.O., Collins, K.L.: Symmetry breaking in graphs. Electron. J. Combin. 3(1), R18 (1996)
Babai, L.: Asymmetric trees with two prescribed degrees. Acta Math. Acad. Sci. Hung. 29, 193–200 (1977)
Boutin, D.L.: Small label classes in 2-distinguishing labelings. Ars Math. Contemp. 1(2), 154–164 (2008)
Boutin, D.L., Imrich, W.: Infinite graphs with finite 2-distinguishing cost. Electron. J. Combin. 21(4), #P4.52 (2014)
Cameron, P.J.: Oligomorphic permutation groups, London Mathematical Society Lecture Notes Series, vol. 152. Cambridge University Press, Cambridge (1990)
Cameron, P.J., Solomon, R., Turull, A.: Chains of subgroups in the symmetric group. J. Algebra 127, 340–352 (1989)
Dixon, J.D., Neumann, P.M., Thomas, S.: Subgroups of small index in infinite symmetric groups. Bull. Lond. Math. Soc. 18, 580–586 (1986)
Evans, D.M.: A note on automorphism groups of countably infinite structures. Arch. Math. 49, 479–483 (1987)
Halin, R.: Automorphisms and endomorphisms of infinite locally finite graphs. In: Abh. Math. Sem. Univ. Hamburg, vol. 39, pp. 251–283 (1973)
Imrich, W., Smith, S.M., Tucker, T.W., Watkins, M.E.: Infinite motion and the distinguishing number of graphs and groups. J. Algebraic Comb. doi:10.1007/s10801-014-0529-2
Kueker, D.W.: Definability, automorphisms, and infinitary languages. In: Barwise, J. (ed.) The Syntax and Semantics of Infinitary Languages, vol. 72. LNM, Berlin (1968)
Reyes, G.E.: Local definability theory. Ann. Math. Logic 1, 95–138 (1970)
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We thank the referee for carefully reviewing the manuscript and for the remarks, which helped to improve the readability of the paper.
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Dedicated to the memory of Rudolf Halin.
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Imrich, W., Smith, S.M. On a theorem of Halin. Abh. Math. Semin. Univ. Hambg. 87, 289–297 (2017). https://doi.org/10.1007/s12188-016-0167-9
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DOI: https://doi.org/10.1007/s12188-016-0167-9