Hypergraph Extension Of The Alon-Tarsi List Coloring Theorem

An Eulerian subgraph in a digraph is a subgraph in which indegree equals outdegree at each vertex. Alon and Tarsi [2] 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 Eulerian subgraphs with even-sized and odd-sized edge sets differ. We generalize this theorem to k-uniform hypergraphs, where k is prime. We use a “hypergraph polynomial” and a concept of hypergraph orientation in which a source is chosen from each edge.