# Complete minors in graphs without sparse cuts

## TL;DR

This paper establishes a lower bound on the size of complete minors in graphs with certain degree and expansion properties, extending known results to various classes of graphs including regular, jumbled, and random graphs.

## Contribution

It introduces a new bound on the order of complete minors in graphs without sparse cuts, applicable to regular, jumbled, and random graphs, generalizing previous results.

## Key findings

- Graphs without sparse cuts contain large complete minors.
- The bound applies to regular, jumbled, and random graphs.
- Largest complete minor size is characterized for various graph classes.

## 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 \[ \Omega\left( \sqrt{\frac{n d}{\log d}} \right). \] As a corollary we determine the order of a largest complete minor one can guarantee in $d$-regular graphs for which the second largest eigenvalue is bounded away from $d/2$, in $(d/n, o(d))$-jumbled graphs, and in random $d$-regular graphs, for almost all $d = d(n)$.