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

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.

💡 Research Summary

The paper investigates the Scarf complex—a subcomplex of the minimal free resolution—of several monomial ideals associated with graphs, providing complete characterizations of when these complexes coincide with the minimal free resolutions (i.e., when the ideals are “Scarf”). The three families studied are: (i) square‑free powers of edge ideals, (ii) symbolic powers of edge ideals, and (iii) cover ideals.

Background. For a monomial ideal (I) in a polynomial ring (S=k