Graeco-Latin Squares and a Mistaken Conjecture of Euler∗

Graeco-Latin Squares and a Mistaken Conjecture of Euler

On the Construction of Sets of Mutually Orthogonal Latin Squares and the Falsity of a Conjecture of Eulero)

then it is known, MacNeish [16] and Mann [17] that there exists a set of at least n(v) mutually orthogonal Latin squares (m.o.l.s.) of order v. It seemed plausible that n(v) is also the maximum possible number of m.o.l.s. of order v. This would have implied the correctness of Euler's [13] conjecture about the nonexistence of two orthogonal Latin squares of order v when v = 2 (mod 4), since n(v)...

Clique Facets of the Orthogonal Latin Squares Polytope∗

Since 1782, when Euler addressed the question of existence of a pair of Orthogonal Latin Squares (OLS) by stating his famous conjecture ([8, 9, 13]), these structures have remained an active area of research due to their theoretical properties as well as their applications in a variety of fields. In the current work we consider the polyhedral aspects of OLS. In particular we establish the dimen...

On the relations of various conjectures on Latin squares and straightening coefficients

We discuss the relations among various combinatorial conjectures, to wit: I. A conjecture of J. Dinitz on partial Latin squares, closely related to a conjecture of N. Alon and M. Tarsi on Latin squares. II. A conjecture of the second author on the nonvanishing property of a certain straightening coefficient in the supersymmetric bracket algebra. This conjecture is motivated by the author’s prog...

Lifting Redundancy from Latin Squares to Pandiagonal Latin Squares

In the pandiagonal Latin Square problem, a square grid of size N needs to be filled with N types of objects, so that each column, row, and wrapped around diagonal (both up and down) contains an object of each type. This problem dates back to at least Euler. In its specification as a constraint satisfaction problem, one uses the all different constraint. The known redundancy result about all dif...