The Scarf complex of squarefree powers, symbolic powers of edge ideals, and cover ideals of graphs

Abstract

Every monomial ideal I has a Scarf complex, which is a subcomplex of its minimal free resolution. We say that I is Scarf if its Scarf complex is also its minimal free resolution. In this paper, we fully characterize all pairs (G,n) of a graph G and an integer n such that the squarefree power I(G)[n] or the symbolic power I(G)(n) of the edge ideal I(G) is Scarf. We also determine all graphs G such that its cover ideal J(G) is Scarf, with an explicit description when G is either chordal or bipartite.