Linear f-connectivity guaranteeing Hamiltonicity

Determine whether there exists a linear function f(k) = Θ(k) such that every f-connected graph (i.e., satisfying |A ∩ B| ≥ f(min(|A \ B|, |B \ A|)) for any partition V(G)=A∪B with no edges between A\B and B\A) is Hamiltonian.

Background

Brandt, Broersma, Diestel, and Kriesell introduced f-connectedness and proved Hamiltonicity under superlinear f; subsequent work improved bounds but left open whether linear f suffices.

The present paper’s Theorem 1.4 on C-expanders implies Theorem 1.9, confirming Hamiltonicity for f(k)=Ck, thereby resolving this conjecture. The conjecture is included here as it was explicitly stated historically and is addressed by the paper.