Magic Eulerian Hypergraphs

Motivated by the phenomenon of contextuality in quantum physics we study a particular family of proofs of the Kochen-Specker theorem. A proof is represented by a pair consisting of a hypergraph where each vertex has even degree, called an Eulerian hypergraph, and a labeling of its vertices. If a hypergraph admits a labeling constituting a proof, we say it is magic. We are interested in determining whether a given hypergraph is magic. Working with the duals of Eulerian hypergraphs, we develop the parity minor relation, which allows us to establish a concept of minimality for the magic property. We introduce a splitting operation to show that certain hypergraphs are not magic. We combine the parity minor relation and the splitting operation in order to search for minimally magic hypergraphs whose duals are grafts, hypergraphs with one edge of size four and all other edges of size two.