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

Abstract

Let $G$ be a bipartite graph and $I=I(G)$ be its edge ideal.  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 $\Rees(I)$ 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 $\reg(I^{s+1})=\reg(I^s)+2$ and that for an $s\ge 1$ we have the inequality $\reg(I^s) \le 2s + \MM(G) -  1$.