Graphs where every vertex can be its own color class

Question

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?

Answer 1

For a quick counter example look at odd cycles.

These graphs are called (vertex) critical. You can look into the Hajós construction, which produces larger critical graphs from smaller ones without changing the chromatic number.