Key points are not available for this paper at this time.
In this paper, we prove that the path ideals of both paths and cycles have minimal cellular resolutions. Specifically, these minimal free resolutions coincide with the Barile-Macchia resolutions for paths, and their generalized counterparts for cycles. Our result on cycles marks the first instance in the literature where the minimal free resolution of edge ideals of cycles has been established. Furthermore, edge ideals of cycles represent the first class of ideals that lack a minimal Barile-Macchia resolution, yet have a minimal generalized Barile-Macchia resolution.
Chau et al. (2024) studied this question.