Acharya–Mütze–Verciani connectivity conjecture for cyclically colored triangulations

The Acharya–Mütze–Verciani conjecture concerns connectivity of reconfiguration graphs for cyclically colored triangulations. A June 2024 paper established an important special case; a new August 2026 preprint claims the full stated result and a unified decomposition framework.

Known results

• The twist graph $\mathcal{H}_N$ is connected when $N$ is divisible by $3$ (Theorem 9, 2024).
• The same 2024 paper conjectured Hamiltonicity of $\mathcal{H}_N$ for $N \ge 9$ divisible by $3$, and observed disconnection when $N \equiv 2 \pmod 3$.

August 2026 claimed resolution

The preprint Cyclically Colored Triangulations: Enumeration and Connectedness of Reconfiguration Graphs claims connectivity of the $3$-color twist graph for $k \ge 4$, exceptions $\mathcal{H}_8$ and $\mathcal{H}_{11}$, and the corresponding flip-graph connectivity for $j \ge 4$. It is explicitly unrefereed, so these claims remain unverified.

Current status (as of August 2026): The $N$-divisible-by-$3$ twist-graph case is established, while the broader $k \ge 4$ and $j \ge 4$ claims, including the stated exceptions, are reported only in an unrefereed preprint and remain unverified.