The Dinitz problem solved for rectangles

Abstract

The Dinitz conjecture states that, for each $n$ and for every collection of $n$-element sets $S_{ij}$, an $n\times n$ partial latin square can be found with the $(i,j)$\<th entry taken from $S_{ij}$. The analogous statement for $(n-1)\times n$ rectangles is proven here. The proof uses a recent result by Alon and Tarsi and is given in terms of even and odd orientations of graphs.