# Hamiltonian Graphs

I am trying to decide whether it is possible to have two graphs with the same degree sequence where both are connected, but only one has a Hamiltonian cycle. Can anyone give me an example?

It is obvious to find examples when one is connected and one is disconnected, however I am unable to come up with anything for this case.

The degree sequence $3,3,2,2,2,2$ can be realized by a hexagon with a chord, or by two triangles with a bridge between them. One is Hamiltonian, the other is not.
Take the following two graphs $G$ and $H$ with the same degree sequence 3,3,2,2,2. $H$ is Hamiltonian while $G$ is not.