Montgomery’s Hamiltonicity conjecture for regular sublinear expanders

Montgomery’s conjecture asks whether the relevant regular sublinear expanders universally contain a Hamiltonian cycle. The latest report says this universal statement is false and identifies the log-squared degree scale as an obstruction.

Known results

• Draganić, Montgomery, Munhá Correia, Pokrovskiy, and Sudakov, 2024: every sufficiently strong constant-expansion graph is Hamiltonian.
• Glock, Munhá Correia, and Sudakov, 2023: Hamiltonicity was proved under the weaker pseudorandomness condition $d/\lambda \geq C\log^{1/3} n$.
• Montgomery, 2026 and collaborators: regular $\gamma$-expanders with $d\geq(\gamma^{-1}\log n)^K$ are Hamiltonian when bipartite or $\gamma$-far from bipartite.

August 2026 counterexample claim

The preprint “Small circumference in regular sublinear expanders” reports a strong counterexample to the proposed universal Hamiltonicity statement, showing that the log-squared degree scale is a genuine obstruction. It is unrefereed, and the retrieved sources provide no independent verification.

Current status (as of August 2026): the universal claim is reported as refuted by an unrefereed preprint, while the counterexample remains unverified.