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 ...
"I'm having trouble finding literature on this type of graph"
A good book recommended by universities is Discrete Mathematics and It's Applications by Kenneth H. Rosen. You can find the chapter on graphs (Euler and Hamilton paths) on page 666.