Vertex cover ideals of simplicial complexes

Abstract

Given a simplicial complex $\Delta$, we investigate how to construct a new simplicial complex $\bar{\Delta}$ such that the corresponding monomial ideals satisfy nice algebraic properties. We give a procedure to check the vertex decomposability of an arbitrary hypergraph. As a consequence, we prove that attaching non-pure skeletons at all vertices of a cycle cover of a simplicial complex $\Delta$ results in a simplicial complex $\bar{\Delta}$ such that the associated hypergraph $\mathcal{H}(\bar{\Delta})$ is vertex decomposable. Also, we prove that all symbolic powers of the cover ideal of $\bar{\Delta}$ are componentwise linear. Our work generalizes the earlier known result where non-pure complete graphs were added to all vertices of a cycle cover of a graph.