Syzygies, Betti numbers and regularity of cover ideals of certain multipartite graphs

TL;DR

This paper investigates the algebraic properties such as syzygies, Betti numbers, and regularity of powers of cover ideals for specific classes of graphs, enhancing understanding of their algebraic and combinatorial structure.

Contribution

It provides explicit descriptions of syzygies, Betti numbers, and regularity for powers of cover ideals of certain multipartite graphs, a novel analysis in this area.

Findings

Explicit formulas for Betti numbers of $J_G^s$

Regularity bounds for powers of cover ideals

Characterization of syzygies for specific graph classes

Abstract

Let $G$ be a finite simple graph on $n$ vertices. Let $J_G \subset K[x_1, \ldots, x_n]$ be the cover ideal of $G$. In this article, we obtain syzygies, Betti numbers and Castelnuovo-Mumford regularity of $J_G^s$ for all $s \geq 1$ for certain classes of graphs $G$.