  We define a new combinatorial object, which we call a labeled hypergraph,
uniquely associated to any square-free monomial ideal. We prove several upper
bounds on the regularity of a square-free monomial ideal in terms of simple
combinatorial properties of its labeled hypergraph. We also give specific
formulas for the regularity of square-free monomial ideals with certain labeled
hypergraphs. Furthermore, we prove results in the case of one-dimensional
labeled hypergraphs.
