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

- $\omega(G) < \chi(G)$ (where $\omega(G)$ is the size of the largest clique in $G$), and
- up to isomorphism, the only proper minor $M$ of $G$ such that there is a graph homomorphism $f:G\to M$ is the complete graph $K_n$ where $n=\chi(G)$.

proper minorinstead ofminor. Otherwise, since there is always a homomorphism from $G$ to $G$, your two conditions are clearly incompatible. $\endgroup$