Is there a polynomial algorithm to solve TSP (or Ham Cycle) on an m by n solid grid graph whose points are at unit distance apart? I've heard about Umans and Lenhart research paper but reading such paper is beyond my capability.
This paper is more general than what you're asking for, but should lead you to some answers. It's pretty accessible too:
(Though looking at it now the primary reference I'd have suggested out of that would be the Umans and Lenhart paper...)