For any set $X$, let $[X]^2 = \big\{\{x,y\}: x\neq y \in X\big\}$. 


Is there a finite, simple, undirected, connected graph $G=(V,E)$ with the following properties?


1. There is $\{v, w\}\in [V]^2\setminus E$ such that collapsing $v,w$ increases the chromatic number, but
2. for all $\{a, b\}\in [V]^2\setminus E$ we have $\chi((V,E)) = \chi((V, (E\cup\{a,b\})))$, that is, adding an edge connecting $a$ and $b$ does not increase the chromatic number.



The effects of collapsing vs joining non-adjacent vertices on the chromatic number
**Yes,** such a graph does exist. Let $G$ be obtained from the complete graph $K\_{100}$ by adding two non-adjacent vertices $v$ and $w$ such that $|N\_G(v)|=|N\_G(w)|=50$ and $N\_G(v) \cup N\_G(w)=V(K\_{100})$. Here, $N\_G(v)$ denotes the set of vertices of $G$ which are adjacent to $v$. Then collapsing $v$ and $w$ in $G$ yields $K\_{101}$, which increases the chromatic number. On the other hand, it is easy to see that adding any edge to $G$ does not increase the chromatic number. 


