Dewar's degree-threshold conjecture for globally rigid vertex-transitive graphs

Dewar’s conjecture asks whether sufficiently high degree forces every vertex-transitive graph to be globally rigid in dimension $d$. A January 2026 paper claims an affirmative answer with the optimal threshold.

Known results

• Villányi’s earlier result gave global rigidity for degree at least $\frac{3}{2}d(d+1)$ in connected vertex-transitive graphs.

January 2026 claimed sharp solution

Angelo El Saliby’s paper Highly regular vertex-transitive graphs are globally rigid claims that every connected vertex-transitive graph of degree at least $d(d+1)$ is globally rigid in $\mathbb{R}^d$. It also claims sharpness: for every $d\ge 2$, a connected vertex-transitive graph of degree $d(d+1)-1$ exists that is not globally rigid in $\mathbb{R}^d$.

Current status (as of September 2026): The conjecture has a claimed solution with threshold $f(d)=d(d+1)$ and claimed sharpness, but the January 2026 preprint remains unverified.