“Hamiltonicity of Expanders: Optimal Bounds and Applications”, Nemanja Draganić, Richard Montgomery, David Munhá Correia, Alexey Pokrovskiy, Benny Sudakov2024-02-09 (algorithms, math)⁠:

[media] An n-vertex graph G is a C-expander if |N(X)| ≥ C|X| for every X ⊆ V(G) with |X| < n⁄2 and there is an edge between every two disjoint sets of at least n⁄C vertices.

We show that there is some constant C > 0 for which every C-expander is Hamiltonian. In particular, this implies the well known conjecture of Krivelevich and Sudakov from 2003 on Hamilton cycles in (n, d, λ)-graphs.

This completes a long line of research on the Hamiltonicity of sparse graphs, and has many applications, including to the Hamiltonicity of random Cayley graphs.