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arXiv · 2304.11640

Vertex cover ideals of simplicial complexes

Bijender·Ajay Kumar
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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.

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Bijender, Ajay Kumar. 2023-04-23. Vertex cover ideals of simplicial complexes. https://arxiv.org/abs/2304.11640

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