This is an update to an [older question](https://mathoverflow.net/questions/290069/identifying-two-non-adjacent-vertices-and-the-effect-on-the-hadwiger-number) admitting a trivial example to answer it.


Suppose $G$ is a finite simple graph. Let $h(G)$ denote the *Hadwiger number* of $G$; that is, the maximum $n\in\mathbb{N}$ such that $K\_n$ is a [minor](https://en.wikipedia.org/wiki/Graph_minor) of $G$. 


Given an integer $n \geq 3$, is there a graph $G = (V,E)$ with $\chi(G) = n$ and with the following property?



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> There are two non-adjacent vertices $v, w\in V$ such that when $v$ and $w$ are identified, the Hadwiger number of the resulting graph is smaller than $h(G)$.
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Hadwiger critical graphs of arbitrarily high chromatic number
This is a generalization of Bjørn Kjos-Hanssen's answer to your previous question.


Take $K\_{n+2}$ and remove two edges between two different pairs of vertices, say $(1,2)$ and $(3,4)$ The chromatic number is $n$. The Hadwiger number is $n$ as well. But if we identify 1 and 3, we get $K\_{n+1}$ with Hadwiger number $n+1$.


