A DG algebra structure for the edge ideal of any split graph

Abstract

We construct an explicit DG algebra structure on the minimal free resolution of the quotient of a polynomial ring by the edge ideal of any split graph (that is, a graph whose vertex set is a disjoint union of a clique and an independent set). The algebra structure is based on a simple combinatorial notion, which we call the fixing edge of a pair of edges, that directly captures the structure of the underlying graph. Our construction involves an explicit description of the minimal free resolution of the edge ideal of such a graph, including formulas for both the Betti numbers and the differentials.