Complete Minors in Graphs Without Sparse Cuts

Abstract

Abstract We show that if $G$ is a graph on $n$ vertices, with all degrees comparable to some $d = d(n)$, and without a sparse cut, for a suitably chosen notion of sparseness, then it contains a complete minor of order $$\begin{equation*} \Omega\left( \sqrt{\frac{n d}{\log d}} \right). \end{equation*}$$As a corollary we determine the order of a largest complete minor one can guarantee in $d$-regular graphs for which the 2nd largest eigenvalue is bounded away from $d/2$, in $(d/n, o(d))$-jumbled graphs, and in random $d$-regular graphs, for almost all $d = d(n)$.