Let $G=(V,E)$ be a finite, simple, undirected graph. We say $G$ has the *singleton coloring class property* (SCCP) if for all $v\_0\in V$ there is a vertex coloring $c:V\to\{1,\ldots,\chi(G)\}$ such that no vertex other than $v\_0$ receives the same color as $v\_0$, that is, $c^{-1}\big(\{c(v\_0)\}\big) =\{v\_0\}$.


If a graph $G$ is connected and $G$ has the SCCP, does this imply that $G$ is complete?



Graphs where every vertex can be its own color class
For a quick counter example look at odd cycles. 


These graphs are called [(vertex) critical](https://en.wikipedia.org/wiki/Critical_graph). You can look into the Hajós construction, which produces larger critical graphs from smaller ones without changing the chromatic number.


