Symbolic Powers of Cover Ideals of Graphs and Koszul Property

TL;DR

This paper demonstrates that attaching whiskers to vertices of a cycle cover in a graph ensures all symbolic powers of its cover ideal are Koszul, extending previous results on whisker attachments.

Contribution

It introduces a new graph construction method that guarantees the Koszul property for all symbolic powers of the cover ideal, generalizing earlier work.

Findings

All symbolic powers of the cover ideal are Koszul.

Extension of previous results to cycle covers.

Generalization to partial whisker attachments.

Abstract

We show that attaching a whisker (or a pendant) at the vertices of a cycle cover of a graph results in a new graph with the following property: all symbolic powers of its cover ideal are Koszul or, equivalently, componentwise linear. This extends previous work where the whiskers were added to all vertices or to the vertices of a vertex cover of the graph.