Distinguishing locally finite trees

TL;DR

This paper investigates coloring strategies for finite and infinite trees to maximize fixed vertices under automorphisms, revealing bounds based on tree valence and color count.

Contribution

It introduces a method to color trees with c colors to fix many vertices, improving automorphism fixing beyond existing bounds.

Findings

Existence of c-colorings fixing vertices at certain distances

Improved fixing results in many cases

Bounds depend on tree valence and number of colors

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.