Minimum number of checks

On another forum, I've found a meaningless question leading to an interesting one.

Suppose a 9x9 Sudoku grid is filled with digits from 1 to 9.

At this point, don't suppose any constraint (different values) on rows/columns/blocks (if we assumed all of them, no more test would be needed).

Now suppose one wants to check that the grid is indeed a valid Sudoku solution and one has already checked that the 27 sums for each row/column/block is 45. These 27 checks can't be enough for all the possible grids (otherwise Sudoku would be an integer linear programming problem with 81 unknowns and 27 linear constraints). There must be some counter-example, but I don't know any.

Now the question: what's the minimum number of elementary checks of type xi≠yi (where xi and yi are the digits in different cells in the same row/column/block) necessary to be sure in all possible cases that the grid is a valid Sudoku solution?

To be more precise, all the xi≠yi must be preceded by a universal quantifier: find the minimum n such that for all such (sum-checked) grids and for all n choices of xi≠yi constraints, these additional tests are enough to make sure the grid is a valid Sudoku solution.

(At first, I thought the answer would be almost trivial if we chose existential quantifiers instead: choose the cell-pairs for which the constraint is not satisfied. But the way to the final answer may not be so trivial finally. So, it might be a second question. No time to investigate now.)