Before revision:  We exhibit classes of graphs of bounded degree that are locally Hamiltonian, i. e.  every subgraph induced by the neighbourhood of a small vertex set appears in some Hamiltonian graph , but that are far from being Hamiltonian.
After revision:  We exhibit classes of graphs of bounded degree that are locally Hamiltonian, i. e.locally Hamiltonian every subgraph induced by the neighbourhood of a small vertex set appears in some Hamiltonian graph , but that are far from being Hamiltonian.