A graph $G=(V,E)$ is said to be *vertex-critical* if removing a vertex $v\in V$ reduces the chromatic number $\chi(\cdot)$. *Edge-criticality* is defined in a similar manner. Moreover, $G$ is called *contraction-critical* if contracting any edge reduces the chromatic number.

*Questions.*

1) Are edge- and vertex-criticality equivalent?

2) What is an example of a graph $G$ such that $\omega(G) < \chi(G)$ [where $\omega(G)$ denotes the clique number), and $G$ is vertex-critical, but not contraction-critical?

extremelysmall counterexample: the non-connected graph $\bullet\quad\bullet$ - $\bullet$ has $\chi=2$, is not vertex-critical (delete $\bullet$) butisedge-critical because deleting the edge makes $\chi$ drop from $2$ to $1$, and there is only this edge. $\endgroup$ – Peter Heinig May 29 '17 at 17:52