Source is in Mathworld, but it has no proof. Also, I am not completely sure what "unbalanced" means here. Non-equal number of even and uneven degrees?
What I am interested in, are there some universal properties of (connected) graphs that prevent hamiltonicity? Something to do with number of vertices versus edges, or average degrees or such? Something that is easy to demonstrate?
I will (hopefully) be using them as counterexamples of hamiltonian graphs in my thesis.