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.

References

Conjecture 1.1. There exists C > 0 such that if d ≥ C then every (n,d,λ)-graph is Hamiltonian. λ

— Hamiltonicity of expanders: optimal bounds and applications  (2402.06603 - Draganić et al., 2024) in Conjecture 1.1, Section 1

The preceding discussion raises the natural question of whether all $(n,d,\lambda)$-graphs with $\lambda\leq d-2$ are Hamiltonian. The author is not aware of any counterexamples to such a conjecture, despite extensive search with modern AI tools. This should of course be taken with a grain of salt, as it remains very difficult to argue, in general, that a given graph is not Hamiltonian.

— Hamiltonicity of mildly pseudorandom regular graphs  (2609.35766 - Müyesser, 28 Sep 2026) in Section 1, Introduction

A more plausible conjecture is that a wider separation between $d$ and $\lambda$ should be sufficient for Hamiltonicity. For any $\delta\in(0,1)$, there exists $d_0$ so that for every $d\geq d_0$, every $(n,d,\lambda)$-graph with $\lambda\leq (1-\delta)d$ is Hamiltonian.

— Hamiltonicity of mildly pseudorandom regular graphs  (2609.35766 - Müyesser, 28 Sep 2026) in Section 1, Introduction, Conjecture (Folklore), labeled Conjecture \ref{conj:main}