Transversals in generalized Latin squares

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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 ) ≤ n 2 /2 if n is large enough. In this note we prove that l ( n ) is bounded from above by 0.75 n 2 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 2 n admits a multicolored 1 -factor.