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

TL;DR

This paper explores the algebraic and combinatorial properties of the Rees algebra of edge ideals of bipartite graphs, providing formulas for regularity and Gr"obner bases, and analyzing the regularity of their powers.

Contribution

It offers explicit descriptions of the Rees algebra's regularity and Gr"obner basis in terms of graph combinatorics, and establishes a linear regularity growth pattern for powers of the ideal.

Findings

Regularity of the Rees algebra is computed using graph combinatorics.

Universal Gr"obner basis of defining equations is characterized combinatorially.

Regularity of powers of the edge ideal increases linearly after a certain threshold.

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 $\text{reg}(I^{s+1})=\text{reg}(I^s)+2$.