Well I'm not sure exactly what the question is trying to test. Depending on the specific reaction conditions I suppose several things could happen.
First off, let's assume it's under acidic conditions (since E1 is an acid mechanism). Either oxygen in red can get protonated, thus becoming a good leaving group, and leave (as either water if the hydroxyl is protonated or as ethanol if the ether is protonated) to form a tertiary carbocation. Which will get protonated you ask? Well it's all in equilibrium so the answer is really that both are getting protonated, deprotonated, reprotonated, etc... H+ is shuffling around.
If we pretend that only the ether gets protonated and leaves then we can eliminate an H to form an enol, which will of course convert to the keto form to give cyclohexanone.
If we pretend that only the hydroxyl gets protonated and leaves then we can again eliminate an H, but this time we form an enol ether. Enol ethers can be isolated, but they can also be hydrolyzed in acid to carbonyls. Thus, depending on the strength of the acid we have in the reaction, this enol ether may be further hydrolyzed to again give a ketone (cyclohexanone).