Circle graphs and the automorphism group of the circle

TL;DR

The paper establishes a precise link between the automorphism group of the circle and a natural circle-graph, proving $\mathrm{Aut}(\mathcal{C}) \cong \mathrm{Aut}(\mathbb{S}^1)$ where $\mathcal{C}$ is the intersection graph of chords of $\mathbb{S}^1$. It further shows that the countable subgraph $\mathcal{C}_{\mathbb{Q}}$ is strongly universal for countable circle graphs and invariant under local complementation, a rare property shared only with $K_2$ and the Rado graph. The work blends circle-geometry, infinite graph theory, and vertex-minor concepts (Bouchet) to extend finite-characterizations to the countable setting and connects to Ivanov-type metaconjectures. It also raises open questions about higher-dimensional analogues via $d$-sphere graphs and about the uniqueness of invariant universal circle graphs.

Abstract

We prove that $Aut({\mathbb S}^1)$ coincides with the automorphism group of the \emph{circle graph} $\mathcal{C}$, i.e. the intersection graph of the family of chords of ${\mathbb S}^1$. We prove that the countable subgraph of $\mathcal{C}$ induced by the rational chords is a strongly universal element of the family of circle graphs, and that it is invariant under local complementation. The only other known connected graphs that have the latter property are $K_2$ and the Rado graph.

Figures (2)

• Figure 1: A contradiction to mapping non-incident chords $C,D$ to incident ones in the proof of Lemma \ref{['lem bc']}.
• Figure 2: Defining the injection $o_q$ in the proof of Theorem \ref{['thm locom']}.

Theorems & Definitions (14)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Proposition 2.1
• proof
• Lemma 3.1
• proof
• Remark 3.2
• proof : Proof of Theorem \ref{['thm aut']}
• Corollary 3.3