## Abstract

For every r∈N, let θ_{r} denote the graph with two vertices and r parallel edges. The θ_{r}-girth of a graph G is the minimum number of edges of a subgraph of G that can be contracted to θ_{r}. This notion generalizes the usual concept of girth which corresponds to the case r=2. In Kühn and Osthus (2003), Kühn and Osthus showed that graphs of sufficiently large minimum degree contain clique-minors whose order is an exponential function of their girth. We extend this result for the case of θ_{r}-girth and we show that the minimum degree can be replaced by some connectivity measurement. As an application of our results, we prove that, for every fixed r, graphs excluding as a minor the disjoint union of kθ_{r}’s have treewidth O(k⋅logk).

Original language | English (US) |
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Pages (from-to) | 106-121 |

Number of pages | 16 |

Journal | European Journal of Combinatorics |

Volume | 65 |

DOIs | |

State | Published - Oct 2017 |

## ASJC Scopus subject areas

- Discrete Mathematics and Combinatorics

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