Minors in graphs of large θ_r-girth

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).