Depth and Stanley depth of powers of the edge ideals of some caterpillar and lobster trees

TL;DR

This paper establishes sharper lower bounds for the depth and Stanley depth of powers of edge ideals associated with certain caterpillar and lobster trees, advancing understanding in combinatorial commutative algebra.

Contribution

It provides new, significantly improved lower bounds for the depth and Stanley depth of powers of edge ideals of specific tree classes.

Findings

Sharper lower bounds for depth of $S/I^t$

Enhanced Stanley depth bounds for these ideals

Applicable to caterpillar and lobster trees

Abstract

Let $S$ be a ring of polynomials in finitely many variables over a field. In this paper we give lower bounds for depth and Stanley depth of modules of the type $S/I^t$ for $t\geq1$, where $I$ is the edge ideal of some caterpillar and lobster trees. These new bounds are much sharper than the existing bounds for the classes of ideals we considered.