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?



Concepts of criticality in graph theory
Concerning your first question, every edge-critical graph without isolated vertices must be vertex-critical, but not vice versa. For instance, the complement of a $7$-cycle is vertex-critical but not edge-critical.


Concerning your second question, every vertex-critical graph must be contraction-critical as well. Suppose we are contracting an edge $uv$ of a $k$-vertex-critical graph $G$. Since $G$ is $k$-vertex-critical, there is a $k$-colouring $c$ of $G$ in which $u$ is the only vertex coloured $k$. This is also a proper colouring of the graph with the edge $uv$ contracted.


