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

  • Authors:

  • Stephan Brandt;Hajo Broersma;Reinhard Diestel;Matthias Kriesell

  • Affiliations:

  • Fak. Mathematik & Naturwissenschaften, Technische Universitäät Ilmenau, 100565, 98684, Ilmenau, Germany;Department of Computer Science, Science Labs, University of Durham, 100565, South Road, 98684, DH1 3LE Durham, United Kingdom;Mathematisches Seminar, Universität Hamburg, 100565, Bundesstr. 55, 20146, Hamburg, Germany;Mathematisches Seminar, Universität Hamburg, 100565, Bundesstr. 55, 20146, Hamburg, Germany

  • Venue:
  • Combinatorica
  • Year:
  • 2006

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Abstract

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.