# Elementary Results on Forbidden Minors

## TL;DR

This paper explores the theory of forbidden minors in graphs, proves Wagner's Theorem via Kuratowski's Theorem, and discusses how forbidding certain minors constrains the chromatic number, relating to Hadwiger's Conjecture.

## Contribution

It provides elementary proofs of key theorems and establishes connections between forbidden minors and graph coloring, advancing understanding of graph minor theory.

## Key findings

- Proved Wagner's Theorem using Kuratowski's Theorem
- Linked forbidden minors to bounds on chromatic number
- Discussed implications for Hadwiger's Conjecture

## Abstract

We start by building up some theory to state Wagner's Theorem, and then prove it using Kuratowski's Theorem, a proof of which is found in Diester (2000). Following this, we establish some connections between the chromatic number of a graph and some of its forbidden minors. The idea is that if we forbid $G$ to have certain graphs as a minor, then the chromatic number of $G$ cannot be too large. Intuitively, this makes sense: if we disallow $G$ from having too many edges, then this makes it easier to colour the graph with fewer colours; we will of course make this precise. We close by explaining how this all relates to Hadwiger's Conjecture.