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Covers and partial transversals of Latin squares

Lunedì 11 giugno 2018, ore 14:30, aula M1.4, edificio Matematica, Dipartimento FIM, Modena

Relatore: Trent G. Marbach (Nankai University, China)

Abstract: The topic of transversals within Latin squares became of interest through the study of mutually orthogonal Latin squares (MOLS). The connection to such useful objects is strong, as a pair of Latin squares are a pair of MOLS if and only if each of the Latin squares decomposes into transversals. Along with this connection to MOLS, transversals have been of interest in the literature recently and have had a number of papers study them in their own right. It is well known that some Latin squares contain no transversals, but what is not so clear is how close a Latin square is to containing a transversal. Previously, this problem has been studied by studying partial transversals of Latin squares and observing how large we can make such a partial transversal. In this presentation, we will discuss an alternate substructure of a Latin square that can also be considered to be close to a transversal, which we call a cover.
A cover of a Latin square is a subset of entries of the Latin square such that each row, column, and symbol is represented at least once in the set of entries. We will present the results we have found regarding covers of minimum size within a Latin square, which shows a kind of duality between covers and partial transversals. We also will present work on minimal covers, which shows a clear distinctiveness between covers and partial transversals. After this, we will demonstrate a few other results of interest on this topic.

[Ultimo aggiornamento: 25/05/2018 12:26:44]