Quasi-transversal in Latin Squares

TL;DR

This paper explores the properties of transversals in Latin squares, establishes conditions for orthogonal mates, and proves Rodney's conjecture for certain graphs, advancing understanding in combinatorics and Latin square theory.

Contribution

It introduces the concept of quasi-transversal, provides an equivalent condition for orthogonal mates, and proves Rodney's conjecture for a specific family of graphs.

Findings

Established a relation between transversals and Latin square graphs

Provided an equivalent condition for Latin squares to have orthogonal mates

Proved Rodney's conjecture for a family of graphs

Abstract

In this paper, we first present the relation between a transversal in a Latin square with some concepts in its Latin square graph, and give an equivalent condition for a Latin square has an orthogonal mate. The most famous open problem involving Combinatorics is to find maximum number of disjoint transversals in a Latin square. So finding some family of decomposable Latin squares into disjoint transversals is our next aim. In the next section, we give an equivalent statement of a conjecture which has been attributed to Brualdi, Stein and Ryser by the concept of quasi-transversal. Finally, we prove the truth of the Rodney's conjecture for a family of graphs.