TITLE: Visit all houses in $m \times n$ town
QUESTION [5 upvotes]: Suppose a town is an $m \times n$ grid of houses, how many ways is there to visit every house exactly once, if one is only allowed to visit one of the (max 4) neighbouring houses in 1 step?
How many if one requires the starting and ending path to be a house on the edge of town?
And how many if one is allowed to visit any of 8 neighbouring houses? (Path should not cross itself.)

REPLY [3 votes]: No closed formula is known for the number of Hamiltonian paths or cycles on a rectangular lattice.
The following entries in OEIS are relevant to your questions: A003763 (cycles on $2n\times 2n$ lattice), A120443 (paths on $n \times n$ lattice), A140519 (king tours on $n \times n$ board) and A140521 (directed king tours on $n \times n$ board).
Here are a few papers that contain research related to this question: 
1981,
1990,
1994,
1997 and
2007.