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.