Depth and Stanley depth of the edge ideals of the powers of paths and cycles

TL;DR

This paper calculates the depth and Stanley depth of edge ideals of powers of paths and cycles, providing explicit formulas and bounds, and confirms a conjecture by Herzog for these cases.

Contribution

It offers explicit formulas for depth and Stanley depth of powers of path and cycle edge ideals, and proves Herzog's conjecture for these ideals.

Findings

Depth and Stanley depth are equal for powers of paths.

Explicit formulas for depth and Stanley depth of path and cycle edge ideals.

Confirmed Herzog's conjecture for these ideals.

Abstract

Let $k$ be a positive integer. We compute depth and Stanley depth of the quotient ring of the edge ideal associated to the $k^{th}$ power of a path on $n$ vertices. We show that both depth and Stanley depth have the same values and can be given in terms of $k$ and $n$. For $n\geq 2k+1$, let $n\equiv 0,k+1,k+2,\dots, n-1(\mod(2k+1))$. Then we give values of depth and Stanley depth of the quotient ring of the edge ideal associated to the $k^{th}$ power of a cycle on $n$ vertices and tight bounds otherwise, in terms of $n$ and $k$. We also compute lower bounds for the Stanley depth of the edge ideals associated to the $k^{th}$ power of a path and a cycle and prove a conjecture of Herzog presented in \cite{HS} for these ideals.