  Hall's Condition is a necessary condition for a partial latin square to be
completable. Hilton and Johnson showed that for a partial latin square whose
filled cells form a rectangle, Hall's Condition is equivalent to Ryser's
Condition, which is a necessary and sufficient condition for completability.
  We give what could be regarded as an extension of Ryser's Theorem, by showing
that for a partial latin square whose filled cells form a rectangle, where
there is at most one empty cell in each column of the rectangle, Hall's
Condition is a necessary and sufficient condition for completability.
  It is well-known that the problem of deciding whether a partial latin square
is completable is NP-complete. We show that the problem of deciding whether a
partial latin square that is promised to satisfy Hall's Condition is
completable is NP-hard.
