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arXiv · 2109.14121

Induced matchings and the v-number of graded ideals

Gonzalo Grisalde·Enrique Reyes·Rafael H. Villarreal
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Abstract

We give a formula for the v-number of a graded ideal that can be used to compute this number. Then we show that for the edge ideal $I(G)$ of a graph $G$ the induced matching number of $G$ is an upper bound for the v-number of $I(G)$ when $G$ is very well-covered, or $G$ has a simplicial partition, or $G$ is well-covered connected and contain neither $4$- nor $5$-cycles. In all these cases the v-number of $I(G)$ is a lower bound for the regularity of the edge ring of $G$. We classify when the upper bound holds when $G$ is a cycle, and classify when all vertices of a graph are shedding vertices to gain insight on $W_2$-graphs.

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Gonzalo Grisalde, Enrique Reyes, Rafael H. Villarreal. 2021-09-29. Induced matchings and the v-number of graded ideals. https://arxiv.org/abs/2109.14121

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