TITLE: Efficient Hamiltonian cycle algorithms for graph classes
QUESTION [7 upvotes]: Generally speaking, finding a Hamiltonian cycle is NP-Hard and so tough. But if $G=L(H)$ is the line graph of $H$, then we can reduce the problem of finding a Hamiltonian cycle in $G$ to finding an Euler tour of $H$, which is easy.
Question is: are there similar shortcuts known for other graph classes?

REPLY [4 votes]: In fact, it can be proved that Hamiltonian Cycle remains $\mathsf{NP}$-hard even for a very restricted subclass of line graphs: line graphs of $1$-subdivisions of planar cubic bipartite graphs. These graphs are cubic. Not surprisingly the same holds if we pass to $4$-regular line graphs: Hamiltonian Cycle remains $\mathsf{NP}$-hard for line graphs of planar cubic bipartite graphs. 
Both results can be obtained by reductions from Hamiltonian Cycle restricted to planar cubic bipartite graphs:
T. Akiyama, T. Nishizeki, N. Saito, NP-Completeness of the Hamiltonian Cycle Problem for Bipartite Graphs, Journal of
Information Processing 3 (1980) 73-76.