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 μ n = 3(n + 1/2 − √ 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 μ n . (4) Every Latin square of order n has a minimal cover of a size which is asymptotically equal to μ n . (5) If 1 k n/2 and n 5 then there is a Latin square of order n with a maximal partial transversal of size n − k. (6) For any ε > 0, asymptotically almost all Latin squares have no maximal partial transversal of size less than n − n 2/3+ε .

Balanced Equi-$n$-Squares

“…The same problem for Latin squares [15] is yet to be resolved. The current best answer to "how close we can get" to a transversal in a Latin square is by Keevash, Pokrovskiy, Sudakov and Yepremyan [23], improving older results by Hatami and Shor [21,28] (see also [6]); similar questions arise for the aforementioned generalized transversals, which is a possible future research direction.…”

“…Along these lines, the existence of a Latin square that decomposes into d × (n/d) submatrices containing all n symbols was resolved in [11]: it is possible for all divisors d of n. In [6], the authors observe that all Latin squares of order n have an O(n 1/2+ε ) × O(n 1/2+ε ) submatrix containing n−O(n 1/2+ε ) distinct symbols, which raised the existence problem for Latin squares of order n 2 which cannot decompose into n × n submatrices which contain all n symbols.…”

Small Latin arrays have a near transversal

“…It is not hard to see that a maximal partial transversal of length ℓ in a Latin square of order n must satisfy n n 2 ∕ ⩽ ℓ ⩽ . In [6] it was shown that for n 5 ⩾ all values of ℓ in this range are achieved. Then Evans [14] constructed an infinite family of Latin squares which simultaneously have maximal partial transversals of each of the permissible lengths.…”

Pursuit-evasion games on latin square graphs

“…Proof. In [12], it was shown that there exists a partial transversal of cardinality n − O log n log log n , and [3] it was shown that a partial transversal of cardinality n − d can be used to construct a cover of cardinality n + d/2. Therefore, there exists a cover of cardinality n + O log n log log n in L. The proof now follows from Theorem 4.2.…”