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Article Dans Une Revue European Journal of Combinatorics Année : 2017

Minors in graphs of large θr-girth

Résumé

For every $r \in \mathbb{N}$, let $\theta_r$ denote the graph with two vertices and $r$ parallel edges. The \emph{$\theta_r$-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 {[Minors in graphs of large girth, {\em Random Structures \& Algorithms}, 22(2):213--225, 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 $\theta_{r}$-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 $O(k\cdot \log k)$.
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Dates et versions

hal-01218519 , version 1 (21-10-2015)
hal-01218519 , version 2 (13-11-2017)

Identifiants

Citer

Dimitris Chatzidimitriou, Jean-Florent Raymond, Ignasi Sau, Dimitrios M. Thilikos. Minors in graphs of large θr-girth. European Journal of Combinatorics, 2017, 65, pp.106-121. ⟨10.1016/j.ejc.2017.04.011⟩. ⟨hal-01218519v2⟩
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