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arXiv · 1801.06731

Regularity and Gr\"obner bases of the Rees algebra of edge ideals of bipartite graphs

Yairon Cid-Ruiz
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Abstract

Let $G$ be a bipartite graph and $I=I(G)$ be its edge ideal. The aim of this note is to investigate different aspects of the Rees algebra $\mathcal{R}(I)$ of $I$. We compute its regularity and the universal Gr\"obner basis of its defining equations; interestingly, both of them are described in terms of the combinatorics of $G$. We apply these ideas to study the regularity of the powers of $I$. For any $s \ge \text{match}(G)+\lvert E(G) \rvert +1$ we prove that $\text{reg}(I^{s+1})=\text{reg}(I^s)+2$.

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Yairon Cid-Ruiz. 2018-01-20. Regularity and Gr\"obner bases of the Rees algebra of edge ideals of bipartite graphs. https://arxiv.org/abs/1801.06731

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