Transversals in generalized Latin squares

János Barát, Zoltán Lóránt Nagy

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 Kn, 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 Kn, 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 K2n admits a multicolored 1-factor.


Keywords


Latin squares, transversals, anti-Ramsey problems, Lovász local lemma

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DOI: https://doi.org/10.26493/1855-3974.1316.2d2

ISSN: 1855-3974

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