# A lower bound on the average degree forcing a minor

## TL;DR

This paper establishes a lower bound on the average degree needed in a graph to guarantee the presence of a given minor, extending previous results and showing the optimality of certain bounds for sparse graphs.

## Contribution

It provides a new lower bound on the average degree that forces a graph to contain a specific minor, generalizing earlier results and confirming the bounds are tight up to a constant.

## Key findings

- Constructs graphs with high average degree avoiding certain minors
- Extends previous bounds for complete and dense graphs
- Shows bounds are tight up to a constant factor

## Abstract

We show that for sufficiently large $d$ and for $t\geq d+1$, there is a graph $G$ with average degree $(1-\varepsilon)\lambda t \sqrt{\ln d}$ such that almost every graph $H$ with $t$ vertices and average degree $d$ is not a minor of $G$, where $\lambda=0.63817\dots$ is an explicitly defined constant. This generalises analogous results for complete graphs by Thomason (2001) and for general dense graphs by Myers and Thomason (2005). It also shows that an upper bound for sparse graphs by Reed and Wood (2016) is best possible up to a constant factor.