The Herschel Graph is richly connected; all vertices have degree 3 or 4. One would intuitively expect the Herschel Graph to be Hamiltonian. However, the Herschel Graph is the smallest, non-Hamiltonian, polyhedral graph.
The algorithm can quickly find a Knight's Tour on a chessboard with cells labelled 1-8. So vertex 11 means cell ( x=1,y=1 ); vertex 23 means cell ( x=2,y=3 ), and so on. The Knight's Tour is represented by an ordered list of vertices which form a Hamiltonian Cycle.