On Null 3-Hypergraphs

Given a 3-uniform hypergraph H consisting of a set V of vertices, and T subset of ((V)(3)) triples, a null labelling is an assignment of +/- 1 to the triples such that each vertex is contained in an equal number of triples labelled +1 and -1. Thus, the signed degree of each vertex is zero. A necessary condition for a null labelling is that the degree of every vertex of H is even. The Null Labelling Problem is to determine whether H has a null labelling. It is proved that this problem is NP-complete. Computer enumerations suggest that most hypergraphs which satisfy the necessary condition do have a null labelling. Some constructions are given which produce hypergraphs satisfying the necessary condition, but which do not have a null labelling. A self complementary 3-hypergraph with this property is also constructed. (C) 2020 Elsevier B.V. All rights reserved.