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


1. $\omega(G) < \chi(G)$ (where $\omega(G)$ is the size of the largest clique in $G$), and
2. 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 $n=\chi(G)$.



Graph homomorphism to minor
I think a 5-cycle meets your needs.


Clique number 2, chromatic number 3. Check.


(It does have a homomorphism to itself, but I assume you meant proper minor.)


It has no homomorphism to a 4-cycle or a 4-path (the two possible minors with 4 edges) and so that leaves a hom to a triangle. Check.


