Hamiltonicity of mildly pseudorandom regular graphs

The problem concerns conditions forcing a Hamilton cycle in regular graphs whose nontrivial eigenvalues are small. The latest claim targets graphs satisfying $\lambda\leq(1-\delta)d$ with degree substantially above $\delta^{-6}(\log n)^3$.

Known results

• Ferber, Han, Mao, and Vershynin: Hamiltonicity when $d\geq(\log n)^6$ and $\lambda\leq cd$ for a sufficiently small absolute constant $c$.
• Glock, Correia, and Sudakov: Hamiltonicity when $d/\lambda\geq C(\log n)^{1/3}$, and for polynomial degree with constant spectral ratio.

September 28, 2026 claimed threshold improvement

Alp Müyesser's new unrefereed preprint is reported to prove Hamiltonicity for $(n,d,\lambda)$-graphs whenever $\lambda\leq(1-\delta)d$ and $d\gg\delta^{-6}(\log n)^3$. If correct, this lowers the degree requirement in the mildly pseudorandom regime; the claim is not independently verified.

Current status (as of September 2026): The stated theorem is claimed in a new preprint, but remains unverified; no public counterexample or refutation was found.