On quasi-equigenerated and Freiman cover ideals of graphs

TL;DR

This paper characterizes which cover ideals of simple graphs are Freiman, a special class of monomial ideals with a predictable number of generators for their powers, focusing on quasi-equigenerated cases.

Contribution

It provides a characterization of Freiman cover ideals of graphs, linking graph properties to algebraic structures of their ideals.

Findings

Identifies conditions under which graph cover ideals are Freiman.

Establishes a connection between graph structure and algebraic properties.

Provides criteria for Freiman ideals within graph theory context.

Abstract

A quasi-equigenerated monomial ideal $I$ in the polynomial ring $R= k[x_1, \ldots, x_n]$ is a Freiman ideal if $\mu(I^2) = l(I)\mu(I)- \binom{l(I)}{2}$ where $l(I)$ is the analytic spread of $I$ and $\mu(I)$ is the number of minimal generators of $I$. Freiman ideals are special since there exists an exact formula computing the minimal number of generators of any of their powers. In this work we address the question of characterizing which cover ideals of simple graphs are Freiman.