Second powers of cover ideals of paths

TL;DR

This paper proves that the second power of the cover ideal of a path graph has linear quotients and introduces a recursive ordering method that extends to chordal graphs, raising new questions about their powers.

Contribution

It introduces a recursive ordering technique to establish linear quotients for the second power of cover ideals of paths, with a natural extension to chordal graphs.

Findings

Second powers of cover ideals of paths have linear quotients.

The recursive order construction generalizes to chordal graphs.

Raises open questions about powers of cover ideals in chordal graphs.

Abstract

We show that the second power of the cover ideal of a path graph has linear quotients. To prove our result we construct a recursively defined order on the generators of the ideal which yields linear quotients. Our construction has a natural generalization to the larger class of chordal graphs. This generalization allows us to raise some questions that are related to some open problems about powers of cover ideals of chordal graphs.