Covers and partial transversals of Latin squares

Darcy Best,Rebecca J Stones,Trent Marbach,Ian M Wanless

doi:10.1007/s10623-018-0499-9

Darcy Best, Rebecca J Stones + Show 2 more

Open Access

https://doi.org/10.1007/s10623-018-0499-9

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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 }.

Highlights

A Latin square of order n is an n × n matrix containing n symbols such that each row and each column contains one copy of each symbol
We have introduced covers of Latin squares with the aim of using them to understand partial transversals better, focusing primarily on topics relating to extremal sizes
We found that some properties of covers have analogous properties for partial transversals, while others do not

Summary

Introduction

A Latin square of order n is an n × n matrix containing n symbols such that each row and each column contains one copy of each symbol. A partial transversal of deficit d is an (n − d)-subset of E(L) in which every line is represented at most once. A Latin square L of order n is equivalent to a tripartite 3-uniform hypergraph with n vertices in each part (corresponding respectively to rows, columns and symbols) and n 2 hyperedges (corresponding to the entries of L). In this framework, a cover of L is precisely an edge cover (a set of hyperedges whose union covers the vertex set) of this hypergraph.

Latin square graph

Covers and partial transversals

Average number of covers

The maximal partial transversal highlighted in the Latin square

Large minimal covers

Concluding remarks