## A reformulation of Hadwiger’s conjecture

Let $G = (V,E)$ be a finite, simple, undirected graph. Hadwiger’s conjecture states that

if $n = \chi(G)$ then $K_n$ is a minor of $G$.

Let’s call that statement (Hadw). We show that it is equivalent to the following statement:

(Hadw’): If $G$ is not a complete graph, then there is a minor $M$ of $G$ such that

1. $M \not \cong G$, and
2. $\chi(M) = \chi(G)$.

It is clear that (Hadw) implies (Hadw’). For the other direction, take any finite graph $G$ and let $n = \chi(G)$. If $G$ is complete, we are done. If not, use (Hadw’) and let $M_1$ be a proper minor such that $\chi(M_1) = n$. If $M_1$ is complete, we are done, otherwise use (Hadw’) again to get a proper minor $M_2$ of $M_1$ with $\chi(M_2) =n$. And so on. Since $G$ is finite, this procedure is bound to end at $M_k$ for some $k\in\mathbb{N}$. We
have $\chi(M_k) = n$, and since the procedure ended at $k$, the graph $M_k$ must be complete. So we get statement (Hadw).