Krivelevich–Sudakov spectral Hamiltonicity conjecture

Determine whether there exists an absolute constant C > 0 such that every d-regular n-vertex (n,d,λ)-graph with spectral ratio d/λ ≥ C is Hamiltonian.

Background

An (n,d,λ)-graph is a d-regular graph on n vertices whose second-largest eigenvalue in absolute value is at most λ, capturing a standard notion of spectral pseudorandomness. Krivelevich and Sudakov (2003) conjectured that a constant spectral gap (i.e., d/λ bounded below by an absolute constant) suffices to guarantee Hamiltonicity.

The paper proves Theorem 1.5, showing that a constant bound on d/λ indeed implies Hamiltonicity, thereby resolving the conjecture. This conjecture historically guided research on Hamiltonicity in sparse pseudorandom graphs and motivated numerous applications.