  Finding Hamitonian Cycles in square grid graphs is a well studied and
important questions. More recent work has extended these results to triangular
and hexagonal grids, as well as further restricted versions. In this paper, we
examine a class of more complex grids, as well as looking at the problem with
restricted types of paths. We investigate the hardness of Hamiltonian cycle
problem in grid graphs of semiregular tessellations. We give NP-hardness
reductions for finding Hamiltonian paths in grid graphs based on all eight of
the semiregular tessilations. Next, we investigate variations on the problem of
finding Hamiltonian Paths in grid graphs when the path is forced to turn at
every vertex. We give a polynomial time algorithm for deciding if a square grid
graph admits a Hamiltonian cycle which turns at every vertex. We then show
deciding if cubic grid graphs, even if the height is restricted to $2$, admit a
Hamiltonian cycle is NP-complete.
