Every Latin square corresponds to a directed acyclic graph with a lattice arrangement, and whose edges indicate order (<). For example:
(source: enjoysudoku.com)
I'm interested in determining which Latin squares have unique orderings, that is, whose corresponding DAG permits no other valid vertex labelling.
A valid labelling maintains uniqueness in rows and columns, and also satisfies the order relationships specified by the edges.
I can't find any references to labelling problems of this particular kind. Does this imply that the DAG structure is not particularly useful for this kind of analysis?
This relates to a study of uniqueness properties wrt Futoshiki puzzles.