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arXiv · 1810.02265

Distinguishing locally finite trees

Svenja Hüning·Wilfried Imrich·Judith Kloas·Hannah Schreiber·Thomas W. Tucker
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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 $\Delta(G)+1$, where $\Delta(G)$ is the maximum valence. Given a finite or infinite tree $T$ of bounded finite valence $k$ and an integer $c$, where $2 \leq c \leq k$, 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 $\lceil\log_ck\rceil$ are fixed by any color preserving automorphism, and that one can do much better in many cases.

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Svenja Hüning, Wilfried Imrich, Judith Kloas, Hannah Schreiber, Thomas W. Tucker. 2018-10-04. Distinguishing locally finite trees. https://arxiv.org/abs/1810.02265

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