A Weak Version of Rota's Bases Conjecture for Odd Dimensions

Abstract

The Alon--Tarsi Latin squares conjecture is extended to odd dimensions by stating it for reduced Latin squares (Latin squares having the identity permutation as their first row and first column). Using a modified version of an identity proved by Onn [Amer. Math. Monthly, 104 (1997), pp. 156--159], we show that the validity of this conjecture implies a weak version of Rota's bases conjecture for odd dimensions, namely that a set of $n$ bases in $\mathbb{R}^n$ has $n-1$ disjoint independent transversals.