A 3-uniform hypergraph H consists of a set V of vertices, and $E\subseteq {V\choose 3}$ triples. Let a null labelling be an assignment of $\pm 1$ to the triples such that each vertex has signed degree equal to zero. A necessary condition is that the degree of every vertex of H is even. The Null Labelling Problem consists in determining whether H has a null labelling. Although the problem is NP-complete, it is of interest to investigate subclasses where the problem turns out to be polynomially solvable. In this study, we define the notion of the $2$-intersection graph related to a 3-uniform hypergraph, and we prove that the existence of a Hamiltonian cycle is sufficient to obtain null labelling in the related hypergraph. The proof we propose provides an efficient way of computing the null labelling.

https://link.springer.com/chapter/10.1007/978-3-030-79987-8_20