Krivelevich–Sudakov conjecture on Hamiltonicity of pseudorandom graphs

Let $G$ be an $(n,d,\lambda)$-graph, meaning a $d$-regular graph on $n$ vertices whose second-largest eigenvalue in absolute value satisfies $|\lambda(G)|\leq\lambda$.

Krivelevich–Sudakov conjecture. There exists $C>0$ such that if

$$\frac{d}{\lambda}\geq C,$$

then every $(n,d,\lambda)$-graph is Hamiltonian.

This conjecture seeks a constant spectral-expansion condition guaranteeing Hamiltonicity, strengthening the previously known bound depending on $n$. It is known in the case $d\geq n^\alpha$ for every fixed $\alpha$, but remains open in general.