Complete minors and average degree -- a short proof

TL;DR

This paper offers a concise, self-contained proof of a classical result linking the average degree of a graph to the size of its complete minor, simplifying the understanding of this fundamental graph theory relationship.

Contribution

It presents a shorter, more accessible proof of the known bound relating average degree and complete minors, enhancing clarity and pedagogical value.

Findings

Graphs with average degree d contain a complete minor of order d/√log d

The proof simplifies previous complex arguments

Reinforces the relationship between average degree and graph minors

Abstract

We provide a short and self-contained proof of the classical result of Kostochka and of Thomason, ensuring that every graph of average degree $d$ has a complete minor of order $d/\sqrt{\log d}$.