Regularity of Powers of Bipartite Graphs

TL;DR

This paper investigates the regularity of powers of edge ideals in bipartite graphs, providing upper bounds, comparisons with associated graphs, and explicit calculations for specific subclasses, advancing understanding of algebraic properties of bipartite graphs.

Contribution

It introduces new upper bounds for the regularity of powers of edge ideals in bipartite graphs and compares properties of related graphs to facilitate explicit calculations.

Findings

Established upper bounds for reg$(I(G)^s)$ in bipartite graphs.

Compared properties of $G$ and $G'$ related to polarization and colon ideals.

Explicitly computed reg$(I(G)^s)$ for certain subclasses of bipartite graphs.

Abstract

Let $G$ be a finite simple graph and $I(G)$ denote the corresponding edge ideal. For all $s \geq 1$, we obtain upper bounds for reg$(I(G)^s)$ for bipartite graphs. We then compare the properties of $G$ and $G'$, where $G'$ is the graph associated with the polarization of the ideal $(I(G)^{s+1} : e_1\cdots e_s)$, where $e_1,\ldots e_s$ are edges of $G$. Using these results, we explicitly compute reg$(I(G)^s)$ for several subclasses of bipartite graphs.