Hypergraph Extension Of The Alon-Tarsi List Coloring Theorem

A circulation in a digraph is a spanning subgraph with indegree equal to outdegree at each vertex. Alon and Tarsi proved that a graph is d-choosable when it has an orientation that has no vertex of outdegree at least d and has the property that the numbers of circulations with even-sized and odd-sized edge sets differ. We generalize this to k-uniform hypergraphs for prime k. We use a “hypergraph polynomial“ and a notion of hypergraph orientation defined by choosing a source vertex from each edge.