Covers and partial transversals of Latin squares

Abstract

We define a cover of a Latin square to be a set of entries that includes at least one representative of each row, column and symbol. A cover is minimal if it does not contain any smaller cover. A partial transversal is a set of entries that includes at most one representative of each row, column and symbol. A partial transversal is maximal if it is not contained in any larger partial transversal. We explore the relationship between covers and partial transversals. We prove the following: (1) The minimum size of a cover in a Latin square of order n is n+a if and only if the maximum size of a partial transversal is either n-2a or n-2a+1. (2) A minimal cover in a Latin square of order n has size at most mu _n=3(n+1/2-sqrt{n+1/4}). (3) There are infinitely many orders n for which there exists a Latin square having a minimal cover of every size from n to mu _n. (4) Every Latin square of order n has a minimal cover of a size which is asymptotically equal to mu _n. (5) If 1leqslant kleqslant n/2 and ngeqslant 5 then there is a Latin square of order n with a maximal partial transversal of size n-k. (6) For any varepsilon >0, asymptotically almost all Latin squares have no maximal partial transversal of size less than n-n^{2/3+varepsilon }.