Depth, Stanley depth and regularity of ideals associated to graphs

TL;DR

This paper investigates algebraic invariants like depth, Stanley depth, and regularity of ideals associated with graphs, establishing bounds based on graph parameters and providing new proofs for known inequalities.

Contribution

It introduces new bounds for Stanley depth of cover ideals and their quotients, and offers an elementary proof for the regularity bound related to the ordered matching number.

Findings

Stanley depth of cover ideals is bounded below by graph parameters.

For bipartite graphs, Stanley depth of powers exceeds depth.

Regularity of the quotient ring is bounded by the ordered matching number.

Abstract

Let $\mathbb{K}$ be a field and $S=\mathbb{K}[x_1,\dots,x_n]$ be the polynomial ring in $n$ variables over $\mathbb{K}$. Let $G$ be a graph with $n$ vertices. Assume that $I=I(G)$ is the edge ideal of $G$ and $J=J(G)$ is its cover ideal. We prove that ${\rm sdepth}(J)\geq n-\nu_{o}(G)$ and ${\rm sdepth}(S/J)\geq n-\nu_{o}(G)-1$, where $\nu_{o}(G)$ is the ordered matching number of $G$. We also prove the inequalities ${\rm sdepth}(J^k)\geq {\rm depth}(J^k)$ and ${\rm sdepth}(S/J^k)\geq {\rm depth}(S/J^k)$, for every integer $k\gg 0$, when $G$ is a bipartite graph. Moreover, we provide an elementary proof for the known inequality ${\rm reg}(S/I)\leq \nu_{o}(G)$.