Scarf complexes of graphs and their powers
Commutative Algebra
2026-03-10 v2 Combinatorics
Abstract
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.
Cite
@article{arxiv.2403.05439,
title = {Scarf complexes of graphs and their powers},
author = {Sara Faridi and Tài Huy Hà and Takayuki Hibi and Susan Morey},
journal= {arXiv preprint arXiv:2403.05439},
year = {2026}
}
Comments
Final version, to appear in Journal of Combinatorial Theory, Series A