Completing partial Latin squares with one filled row, column and symbol

Let P be an n × n partial Latin square every non-empty cell of which lies in a fixed row r , a fixed column c or contains a fixed symbol s . Assume further that s is the symbol of cell ( r , c ) in P . We prove that P is completable to a Latin square if n ≥ 8 and n is divisible by 4 , or n ≤ 7 and n ∉ { 3 , 4 , 5 } . Moreover, we present a polynomial algorithm for the completion of such a partial Latin square.