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? 



Efficient Hamiltonian cycle algorithms for graph classes
The [graph classes webpage](http://www.graphclasses.org/index.html) has a page with a list of some graph classes for which the **[complexity of Hamiltonian Cycle](http://www.graphclasses.org/classes/problem_Hamiltonian_cycle.html)** is known. It currently lists some 400+ graph classes for which the problem is known to be polynomial time solvable.


*Edit: I misread your question as whether there are other graph classes for which Hamiltonian Cycle is polytime solvable, but realized now that that was maybe not what you ment. I will leave this answer nevertheless as it might be of interest. Perhaps more related to your question is that maximum independent set (NP-hard on general graphs) in a line graph corresponds to a maximum matching in its host graph. Hence, independent set is also solvable in polytime on line graphs.*


