Rooted order on minimal generators of powers of some cover ideals

TL;DR

This paper introduces a new rooted order on minimal generators of powers of cover ideals of path graphs, proving linear quotients and extending the concept to chordal graphs, with computational evidence supporting broader applicability.

Contribution

It defines a rooted order on minimal generators of powers of cover ideals of path graphs and extends this concept to chordal graphs, demonstrating linear quotients.

Findings

Powers of cover ideals of path graphs have linear quotients under the rooted order.

Characterization of minimal generators for higher powers based on the second power.

Computational evidence suggests the rooted order may apply to chordal graphs.

Abstract

We define a total order, which we call rooted order, on minimal generating set of $J(P_n)^s$ where $J(P_n)$ is the cover ideal of a path graph on $n$ vertices. We show that each power of a cover ideal of a path has linear quotients with respect to the rooted order. Along the way, we characterize minimal generating set of $J(P_n)^s$ for $s\geq 3$ in terms of minimal generating set of $J(P_n)^2$. We also discuss the extension of the concept of rooted order to chordal graphs. Computational examples suggest that such order gives linear quotients for powers of cover ideals of chordal graphs as well.