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≤(nf(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.

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Correspondence to Stephan Brandt.

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Brandt, S., Broersma, H., Diestel, R. et al. Global Connectivity And Expansion: Long Cycles and Factors In f-Connected Graphs. Combinatorica 26, 17–36 (2006). https://doi.org/10.1007/s00493-006-0002-5

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Mathematics Subject Classification (2000):

  • 05C40
  • 05C38
  • 05C70
  • 05C45