Distinguishing finite and infinite trees of arbitrary cardinality

TL;DR

This paper investigates conditions under which trees and tree-like graphs have vertex colorings that break all non-trivial automorphisms, providing bounds on degrees and counting such colorings for finite and infinite cases.

Contribution

It establishes new bounds on vertex degrees ensuring automorphism-breaking colorings and characterizes the number of such colorings for various classes of trees and graphs.

Findings

For finite trees, elta(T) ^{m(T)/2} for automorphism-breaking colorings.

Infinite trees have 2^{|T|} automorphism-breaking colorings.

Tree-like graphs with degree elta(G) ^{\u001aleph_0} also have 2^{|G|} such colorings.

Abstract

Let $G$ be a finite or infinite graph and $m(G)$ the minimum number of vertices moved by the non-identity automorphisms of $G$. We are interested in bounds on the supremum $\Delta(G)$ of the degrees of the vertices of $G$ that assure the existence of vertex colorings of $G$ with two colors that are preserved only by the identity automorphism, and, in particular, in the number $a(G)$ of such colorings that are mutually inequivalent. For trees $T$ with finite $m(T)$ we obtain the bound $\Delta(T)\leq2^{m(T)/2}$ for the existence of such a coloring, and show that $a(T)= 2^{|T|}$ if $T$ is infinite. Similarly, we prove that $a(G) = 2^{|G|}$ for all tree-like graphs $G$ with $\Delta(G)\le 2^{\aleph_0}$. For rayless or one-ended trees $T$ with arbitrarily large infinite $m(T)$, we prove directly that $a(T)= 2^{|T|}$ if $\Delta(T)\le 2^{m(T)}$.