Let $G = (V,E)$ be a finite, simple, undirected graph. [Hadwiger's conjecture](https://en.wikipedia.org/wiki/Hadwiger_conjecture_(graph_theory)) states that



> (Hadw): $K\_{\chi(G)}$ is a minor of $G$.


It turns out that for finite graphs, (Hadw) is equivalent to the following statement:   
  




>  (Hadw2):   
>   
>  If $G$ is not a complete graph, then there is a minor $M$ of $G$ such that   
>   
> 
> 1. $M \not \cong G$, and
> 2. $\chi(M) = \chi(G)$.
> 
> 
> 


(For an explation of the equivalence of (Hadw) and (Hadw2) in the finite case, see [this](https://dominiczypen.wordpress.com/2015/10/23/a-reformulation-of-hadwigers-conjecture/).)


It is easy to see that (Hadw) fails for graphs with infinite chromatic number: $G:=\bigcup\_{n\in\omega} K\_n$ has chromatic number $\omega$, but $K\_\omega$ is not a minor of $G$.


**Question**: Is (Hadw2) also false for graphs with infinite chromatic number?



Does this version of Hadwiger's conjecture hold for graphs with infinite chromatic number?
First, suppose $G$ has a connected component $C\subseteq G$ with the same chromatic number as $G$. If $C\neq G$, we can take $M=C$. If $G=C$, let $M$ be the subgraph of $G$ obtained by removing all edges involving some fixed vertex $v\in G$.


Now suppose $G$ has no connected component with the same chromatic number as $G$. It follows that we can choose an collection of connected components $C\_i$ of $G$ such that each $C\_i$ has a different chromatic number and $\chi(G)=\sup\_i \chi(C\_i)$. You can now let $M$ be the union of all but one of the $C\_i$.


