Generalized Latin squares of order n with n 2 − 1 distinct elements

Abstract

Let n ≥ 3 be a positive integer. We show that the number of equivalence classes of generalized Latin squares of order n with n2 − 1 distinct elements is 4 if n = 3 and 5 if n ≥ 4. It is also shown that all these squares are embeddable in groups. As an application, we obtain a lower bound for the number of isomorphism classes of certain Eulerian graphs with n2 + 2n − 1 vertices.