### 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.

Original language | English |
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Pages (from-to) | 17-36 |

Number of pages | 20 |

Journal | Combinatorica |

Volume | 26 |

Issue number | 1 |

DOIs | |

Publication status | Published - Feb 2006 |

### Keywords

- MSC-05C70
- MSC-05C38
- MSC-05C40
- MSC-05C45

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## Cite this

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