Hamiltonicity of C-expanders

Determine whether, for every sufficiently large constant C > 0, every n-vertex graph satisfying the C-expansion properties (|N(X)| ≥ C|X| for all X with |X| < n/(2C) and an edge between any two disjoint sets of size at least n/(2C)) is Hamiltonian.

Background

C-expanders formalize vertex expansion and robust edge presence between large sets. This conjecture asks whether sufficiently strong expansion alone forces Hamiltonicity in sparse graphs.

The paper proves Theorem 1.4, establishing the conjecture and providing optimal bounds, with further strengthening to Hamilton-connectedness in the concluding remarks.

References

Conjecture 1.3. For every sufficiently large C > 0, every C-expander is Hamiltonian.

— Hamiltonicity of expanders: optimal bounds and applications  (2402.06603 - Draganić et al., 2024) in Conjecture 1.3, Section 1

It is natural to ask whether the logarithmic-square scale is already sufficient on the positive side.

\begin{problem} For every $\varepsilon>0$, does there exist $C=C(\varepsilon)>0$ such that every sufficiently large $n$-vertex $d$-regular $(\varepsilon,d)$-expander with $d\ge C\log2 n$ contains a Hamilton cycle? \end{problem}

— Small circumference in regular sublinear expanders  (2608.20190 - Chen et al., 20 Aug 2026) in Problem, Section 4 (Concluding remarks)