Rooted order on minimal generators of powers of some cover ideals

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 $s\ge 3$ 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.

16 pages, 3 figures, accepted for publication in Osaka Journal of Mathematics

References in corpus (2)

• Second powers of cover ideals of paths
• On the initial behaviour of the number of generators of powers of monomial ideals