Let's define the following two properties first. A graph $G$ is called dense if it has no minors that have a higher average degree. A $k$-chromatic graph $G$ is called saturated if every minor of $G$ is $k$-colorable, (or not having $K_{k+1}$ as minor if we assume Hadwiger conjecture).
I was trying to came up with properties and examples of dense graphs, and had the conjecture that all dense graphs are also saturated. If someone could come up with a counter example or a proof, or any other interesting property of dense graphs, that would be great.