Given a function f : ℕ→ℝ, call an n-vertex graph f-connected if separating off k vertices requires the deletion of at least f(k) vertices whenever k≤(n−f(k))/2. This is a common generalization of vertex connectivity (when f is constant) and expansion (when f is linear). We show that an f-connected graph contains a cycle of length linear in n if f is any linear function, contains a 1-factor and a 2-factor if f(k)≥2k+1, and contains a Hamilton cycle if f(k)≥2(k+1)2. We conjecture that linear growth of f suffices to imply hamiltonicity.
Brandt, S., Broersma, H., Diestel, R., & Kriesell, M. (2006). Global Connectivity And Expansion: Long Cycles and Factors In f-Connected Graphs. Combinatorica, 26(1), 17-36. https://doi.org/10.1007/s00493-006-0002-5