Global Connectivity And Expansion: Long Cycles and Factors In f-Connected Graphs

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). We conjecture that linear growth of f suffices to imply hamiltonicity.