Elementary Results on Forbidden Minors

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.