Transversals in generalized Latin squares

TL;DR

This paper investigates conditions for the existence of transversals in generalized Latin squares, relating them to edge-colorings of bipartite graphs and proposing bounds and conjectures connected to anti-Ramsey problems.

Contribution

It establishes an upper bound of 0.75n^2 for the function l(n) related to multicolored perfect matchings in edge-colored complete bipartite graphs and connects the problem to anti-Ramsey theory.

Findings

Proved l(n) ≤ 0.75n^2 for n > 1.

Connected the problem to anti-Ramsey problems.

Proposed a new conjecture related to multicolored 1-factors.

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 $l(n)\leq n^2/2$ if $n$ is large enough. In this note we prove that $l(n)$ is bounded from above by $0.75n^2$ 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 $K_{2n}$ admits a multicolored $1$-factor.