Transversals in generalized Latin squares

Abstract

We are seeking a sufficient condition that forces a transversal in a generalized Latin square. A generalized Latin square of order n is equivalent to a proper edge-coloring of K_n, n. A transversal corresponds to a multicolored perfect matching. Akbari and Alipour defined l(n) as the least integer such that every properly edge-colored K_n, n, which contains at least l(n) different colors, admits a multicolored perfect matching. They conjectured that l(n) ≤ n²/2 if n is large enough. In this note we prove that l(n) is bounded from above by 0.75n² if n > 1. We point out a connection to anti-Ramsey problems. We propose a conjecture related to a well-known result by Woolbright and Fu, that every proper edge-coloring of K_2n admits a multicolored 1-factor.