E1 in 6 membered rings
Well, the answer to your question, without any complications, is pretty simple 

Whatever can be forced into planarity in order to bear the $sp^2$ carbocation will undergo $E1$, this can be easily achieved in cyclohexane, as it can easily have a twisted conformation which is relatively less stable, but yet can bear the carbocation. Same is not the case with bridgehead carbons.