Hypergraphs with high projective dimension and 1-dimensional Hypergraphs

TL;DR

This paper characterizes conditions for the projective dimension of square-free monomial ideals linked to hypergraphs, especially 1-dimensional ones with limited cycles, and provides algorithms for their computation.

Contribution

It offers necessary and sufficient conditions for projective dimension and an explicit algorithm for 1-dimensional hypergraphs with at most one cycle per component.

Findings

Characterization of projective dimension for hypergraphs with specific properties

Explicit algorithm for computing projective dimension of 1-dimensional hypergraphs

Application to hypergraphs with spanning Ferrers graphs

Abstract

We prove a sufficient and a necessary condition for a square-free monomial ideal $J$ associated to a (dual) hypergraph to have projective dimension equal to the minimal number of generators of $J$ minus 2. We also provide an effective explicit procedure to compute the projective dimension of 1-dimensional hypergraphs $\mathcal{H}$ when each connected component contains at most one cycle. An algorithm to compute the projective dimension is also included. Applications of these results are given; they include, for instance, computing the projective dimension of monomial ideals whose associated hypergraph has a spanning Ferrers graph.