Distinguishing extension numbers for $\mathbf R^n$ and $S^n$

TL;DR

This paper calculates the distinguishing extension number for certain geometric and combinatorial structures, resolving some conjectures and disproving others, thereby advancing understanding of symmetry-breaking colorings.

Contribution

It explicitly computes the distinguishing extension number for circles and cycle vertices, and provides bounds and counterexamples for Euclidean spaces, addressing open conjectures in the field.

Findings

Computed extension numbers for $S^1$ and cycle vertices.

Proved finiteness of extension number for $ ext{R}^2$ with $SE(2)$.

Disproved conjectures for $ ext{R}^n$ and $S^{n-1}$ with their isometry groups.

Abstract

In the setting of a group $\Gamma$ acting faithfully on a set $X$, a $k$-coloring $c: X\rightarrow \{1, 2, ..., k\}$ is called $\Gamma$-distinguishing if the only element of $\Gamma$ that fixes $c$ is the identity element. The distinguishing number $D_\Gamma(X)$ is the minimum value of $k$ such that a $\Gamma$-distinguishing $k$-coloring of $X$ exists. Now, fixing $k= D_\Gamma(X)$, a subset $W\subset X$ with trivial pointwise stabilizer satisfies the precoloring extension property $P(W)$ if every precoloring $c: X-W\rightarrow \{1, ..., k\}$ can be extended to a $\Gamma$-distinguishing $k$-coloring of $X$. The distinguishing extension number $\text{ext}_D(X, \Gamma)$ is then defined to be the minimum $n$ such that for all applicable $W\subset X$, $|W|\geq n$ implies that $P(W)$ holds. In this paper, we compute $\text{ext}_D(X, \Gamma)$ in two particular instances: when $X = S^1$ is the unit circle and $\Gamma = \text{Isom}(S^1) = O(2)$ is its isometry group, and when $X = V(C_n)$ is the set of vertices of the cycle of order $n$ and $\Gamma = \text{Aut}(C_n) = D_n$, the dihedral group of a regular $n$-gon. This resolves two conjectures of Ferrara, Gethner, Hartke, Stolee, and Wenger. In the case of $X=\mathbf R^2$, we prove that $\text{ext}_D(\mathbf R^2, SE(2))<\infty$, which is consistent with (but does not resolve) another conjecture of Ferrara et al. On the other hand, we also prove that for all $n\geq 3$, $\text{ext}_D(S^{n-1}, O(n)) = \infty$, and for all $n\geq 3$, $\text{ext}_D(\mathbf R^n, E(n))=\infty$, disproving two other conjectures from the same authors.