# Regularity, Rees algebra and Betti numbers of certain cover ideals

## TL;DR

This paper investigates the algebraic properties of certain cover ideals, focusing on their Rees algebra, Betti numbers, and regularity, revealing independence of Betti numbers from tree choices and analyzing powers of these ideals.

## Contribution

It introduces new results on the Betti numbers and regularity of powers of cover ideals, especially for ideals related to graphs and their complements.

## Key findings

- Betti numbers of powers of cover ideals of complement graphs of trees are independent of the tree.
- The paper establishes formulas for the regularity of certain cover ideals.
- It provides insights into the algebraic structure of Rees algebras for specific monomial ideals.

## Abstract

Let $S={\sf k}[X_1,\dots, X_n]$ be a polynomial ring, where ${\sf k}$ is a field. This article deals with the defining ideal of the Rees algebra of squarefree monomial ideal generated in degree $n-2$. As a consequence, we prove that Betti numbers of powers of the cover ideal of the complement graph of a tree do not depend on the choice of tree. Further, we study the regularity and Betti numbers of powers of cover ideals associated to certain graphs.