Depth of the 3-path Ideal of Square of a Path

TL;DR

This paper investigates the algebraic properties of the 3-path ideal of the square of a path graph, specifically its depth and Cohen-Macaulayness, and explores the behavior of the depth of its powers.

Contribution

It provides explicit depth calculations for the 3-path ideal of the square of a path and characterizes when it is Cohen-Macaulay, also analyzing the depth of its powers.

Findings

Depth of the 3-path ideal computed explicitly

Cohen-Macaulay property characterized for specific cases

Limit behavior of depth of powers analyzed

Abstract

In this note, we compute depth of the 3-path ideal of square of a path and show that the 3-path ideal I3(P 2 n) of square of a path graph is Cohen-Macaulay if and only if n = 3 or 4. Also, we consider the limit behavior of depth of powers of the 3-path ideal of square of a path.