Finding a Hamiltonian path in this graph family
The class of graphs that contain a Hamiltonian cycle is known as Hamiltonian graphs. Being Hamiltonian doesn't really imply much: a wide range of well-known graph parameters (diameter, treewidth, chromatic number, ...) can still be unbounded. At least partly, this is the reason there aren't many non-trivial subclasses of Hamiltonian graphs known. 

As an example, every 2-connected $(P_6,K_{1,3})$-free graph is Hamiltonian (see e.g., [1]). Further, [1] also cites some conjectures which claim that every 2-tough graph is Hamiltonian, or that every 4-connected $K_{1,3}$-free graph is Hamiltonian.

On the algorithmic side, even recognizing Hamiltonian graphs is of course well-known to be NP-complete. That is, recognition means the problem of deciding whether a given graph is Hamiltonian or not. Also, because many structural parameters remain unbounded, many classical problems also remain hard for Hamiltonian graphs (as there is no clear algorithmic foothold to exploit).

You might also enjoy Hypohamiltonian graphs, which are "almost Hamiltonian".



[1] R. Faudree, E. Flandrin, Z. Ryjacek. Claw-free graphs - a survey. Discrete Math. 164 87-147 (1997)