Every multigraded free resolution of a monomial ideal I contains the Scarf multidegrees of I. We say I has a Scarf resolution if the Scarf multidegrees are sufficient to describe a minimal free resolution of I. The main question of this paper is which graphs G have edge ideal I(G) with a Scarf resolution? We show that I(G) has a Scarf resolution if and only if G is a gap-free forest. We also classify connected graphs for which all powers of I(G) have Scarf resolutions. Along the way, we give a concrete description of the Scarf complex of any forest. For a general graph, we give a recursive construction for its Scarf complex based on Scarf complexes of induced subgraphs.
