Minors in graphs of large θr-girth

For every r is an element of N, let theta(r) denote the graph with two vertices and r parallel edges. The theta(r)-girth of a graph G is the minimum number of edges of a subgraph of G that can be contracted to theta(r). This notion generalizes the usual concept of girth which corresponds to the case r = 2. In Ktihn and Osthus (2003), Kuhn 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 theta(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 k theta(r)'s have treewidth 0(k . log k). (C) 2017 Elsevier Ltd. All rights reserved.