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Highly regular vertex-transitive graphs are globally rigid

Angelo El Saliby

TL;DR

Global rigidity in higher dimensions for vertex-transitive graphs is addressed. The authors adapt the framework of weakly globally linked pairs, the rigidity matroid, and ordering-induced subgraphs to establish a sharp degree bound. They prove that any connected vertex-transitive graph with degree at least $d(d+1)$ is globally rigid in $\,\mathbb{R}^d$, and provide tightness examples with degree $d(d+1)-1$ that are not globally rigid. This resolves Dewar's conjecture up to the optimal constant and advances the understanding of rigidity in highly symmetric graphs.

Abstract

A graph is said to be globally rigid in $d$-dimensional space if almost all of its embeddings are unique up to isometries. If a graph has enough automorphisms to send any of its vertices into any other, then it is called vertex-transitive. We show that, in any dimension, highly regular vertex-transitive graphs are globally rigid, positively answering a conjecture of Sean Dewar. Furthermore, we construct examples that show that our constant for regularity is best possible.

Highly regular vertex-transitive graphs are globally rigid

TL;DR

Global rigidity in higher dimensions for vertex-transitive graphs is addressed. The authors adapt the framework of weakly globally linked pairs, the rigidity matroid, and ordering-induced subgraphs to establish a sharp degree bound. They prove that any connected vertex-transitive graph with degree at least is globally rigid in , and provide tightness examples with degree that are not globally rigid. This resolves Dewar's conjecture up to the optimal constant and advances the understanding of rigidity in highly symmetric graphs.

Abstract

A graph is said to be globally rigid in -dimensional space if almost all of its embeddings are unique up to isometries. If a graph has enough automorphisms to send any of its vertices into any other, then it is called vertex-transitive. We show that, in any dimension, highly regular vertex-transitive graphs are globally rigid, positively answering a conjecture of Sean Dewar. Furthermore, we construct examples that show that our constant for regularity is best possible.
Paper Structure (5 sections, 13 theorems, 23 equations)

This paper contains 5 sections, 13 theorems, 23 equations.

Table of Contents

  1. Introduction
  2. Weakly globally linked pairs
  3. Weakly globally linked pairs
  4. Ordering-induced subgraphs
  5. Vertex-transitive graphs

Key Result

Theorem 1.1

If $G$ is a connected vertex-transitive graph of degree at least $d(d+1)$, then $G$ is globally rigid in $\mathbb{R}^d$. Moreover, there exist connected vertex-transitive graphs of degree $d(d+1)-1$ that are not globally rigid in $\mathbb{R}^d$.

Theorems & Definitions (25)

  • Conjecture : dewar2023
  • Theorem 1.1
  • Definition 2.1
  • Definition 2.2
  • Theorem 2.3: jordan2024
  • Lemma 2.4
  • Lemma 2.5: villanyi2023
  • Definition 2.6
  • Definition 2.7
  • Lemma 2.8: villanyi2023
  • ...and 15 more