I know that if a graph is Eulerian then there exists an Eulerian cycle that contains all edges of the graph. I also know that if a graph is Hamiltonian then there exists a Hamiltonian cycle that contains all vertices of the graph.
It is easy for me to observe that a Hamiltonian graph may not be Eulerian (because may exist edges not contained in the Hamiltonian cycle). However, I'm a bit confused about the other direction. Is all Eulerian graphs also Hamiltonian?