### Abstract

The distinguishing number D(G) of a graph G is the smallest number of colors that is needed to color the vertices of G such that the only color preserving automorphism is the identity. For infinite graphs D(G) is bounded by the supremum of the valences, and for finite graphs by Δ(G)+1, where Δ(G) is the maximum valence. Given a finite or infinite tree T of bounded finite valence k and an integer c, where 2≤c≤k, we are interested in coloring the vertices of T by c colors, such that every color preserving automorphism fixes as many vertices as possible. In this sense we show that there always exists a c-coloring for which all vertices whose distance from the next leaf is at least ⌈log

_{c}k⌉ are fixed by any color preserving automorphism, and that one can do much better in many cases.Originalsprache | englisch |
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Fachzeitschrift | The Australasian Journal of Combinatorics |

Publikationsstatus | Eingereicht - 2018 |

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## Dieses zitieren

Schreiber, H., Hüning, S., Kloas, J., Imrich, W., & Tucker, T. (2018). Distinguishing locally finite trees. Manuskript zur Veröffentlichung eingereicht.