On quasi-equigenerated and Freiman cover ideals of graphs

Abstract

A quasi-equigenerated monomial ideal I in the polynomial ring $R=k[x_1,\dots ,x_n]$ is a Freiman ideal if $\mu \left(I^2\right)=l\left(I\right)\mu \left(I\right)-\left(\begin{array}{c}l\left(I\right)\\ 2\end{array}\right)$ where l(I) is the analytic spread of I and $\mu \left(I\right)$ 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.

Keywords:

• Cover ideals of graphs
• fiber cone
• Freiman ideals
• monomial ideals

MATHEMATICS SUBJECT CLASSIFICATION:

• 13F20
• 13A30
• 13C05
• 05C25

Acknowledgments

Special thanks are addressed to Professor Alexandra Seceleanu for inviting him to visit University of Nebraska, and for the interesting conversations about the content of this work.

Additional information

Funding

The first author is partially supported by NSF grant DMS-1601024, EPSCoR grant OIA-1557417, and the University of Nebraska – Lincoln. The second author has received financial support from Indam (Istituto Nazionale di Alta Matematica) to spend three months of Spring 2019 at Ohio State University. This article has been carried out during this period of time and the author acknowledges the support of Indam for giving him this opportunity of research.