[Paper Review] On the Stanley depth of edge ideals of line and cyclic graphs

Mircea Cimpoeaş|arXiv (Cornell University)|Nov 3, 2014

Commutative Algebra and Its Applications14 references8 citations

TL;DR

This paper proves that the Stanley conjecture holds for edge ideals of line and cyclic graphs, computing the Stanley depth of the quotient ring $S/J_n$ for cycle graphs: exact values when $n \equiv 0,2 \pmod{3}$, and tight bounds when $n \equiv 1 \pmod{3}$. It also establishes combinatorial bounds for the Stanley depth of quotients of monomial ideals using poset decompositions and generator-based estimates.

ABSTRACT

We prove that the edge ideals of line and cyclic graphs and their quotient rings satisfy the Stanley conjecture. We compute the Stanley depth for the quotient ring of the edge ideal associated to a cycle graph of length $n$, given a precise formula for $n\equiv 0,2 \pmod{3}$ and tight bounds for $n\equiv 1 \pmod{3}$. Also, we give bounds for the Stanley depth of a quotient of two monomial ideals, in combinatorial terms.

Motivation & Objective

• To verify the Stanley conjecture for edge ideals of line and cyclic graphs.
• To compute the Stanley depth of the quotient ring $S/J_n$ associated with the edge ideal of a cycle graph of length $n$.
• To establish combinatorial upper and lower bounds for the Stanley depth of quotients of monomial ideals.
• To provide a method for estimating Stanley depth using poset decompositions and minimal monomial generators.

Proposed method

• Uses the Depth Lemma and inductive arguments to compute the depth of $S/J_n$, showing $\operatorname{depth}(S/J_n) = \left\lceil \frac{n-1}{3} \right\rceil$.
• Applies Stanley decomposition techniques to compute $\operatorname{sdepth}(S/J_n)$ via poset structures on square-free monomials.
• Derives a combinatorial criterion for upper bounds on Stanley depth using coefficients $\alpha_t$ in a recursive formula based on $\beta_t$, the number of independent sets of size $t$ in the cycle.
• Establishes a lower bound for $\operatorname{sdepth}(J/I)$ using the minimal number of generators of $I$ and $J$, leveraging Proposition 2.7 and Theorem 1.4.
• Employs computer-assisted experimentation (via CoCoA) to test conjectures on coefficient non-negativity for $\alpha_t$.
• Uses isomorphisms of quotient rings to reduce the structure of $S/J_n$ to simpler rings like $S_{n-3}/I_{n-3}$ and $S_{n-1}/I_{n-1}$.

Experimental results

Research questions

• RQ1 Does the Stanley conjecture hold for the edge ideal of a cycle graph?
• RQ2 What is the exact value of the Stanley depth of $S/J_n$ for a cycle graph of length $n$?
• RQ3 Can tight combinatorial bounds be established for $\operatorname{sdepth}(S/J_n)$ when $n \equiv 1 \pmod{3}$?
• RQ4 What is the upper bound for the Stanley depth of a quotient of two square-free monomial ideals in terms of poset invariants?
• RQ5 Can a lower bound for $\operatorname{sdepth}(J/I)$ be derived from the number of minimal generators of $I$ and $J$?

Key findings

• The Stanley conjecture holds for the edge ideals of both line and cyclic graphs, as $\operatorname{sdepth}(S/J_n) \geq \operatorname{depth}(S/J_n) = \left\lceil \frac{n-1}{3} \right\rceil$.
• For $n \equiv 0 \pmod{3}$ or $n \equiv 2 \pmod{3}$, $\operatorname{sdepth}(S/J_n) = \left\lceil \frac{n-1}{3} \right\rceil$.
• For $n \equiv 1 \pmod{3}$, $\left\lceil \frac{n-1}{3} \right\rceil \leq \operatorname{sdepth}(S/J_n) \leq \left\lceil \frac{n}{3} \right\rceil$, with exact value $\operatorname{sdepth}(S/J_7) = 2$.
• An upper bound for $\operatorname{sdepth}(J/I)$ is given via the coefficients $\alpha_t$ in a recursive formula based on the number of independent sets in the associated poset.
• A lower bound for $\operatorname{sdepth}(J/I)$ is established as $n - p - \left\lfloor \frac{q - r}{2} \right\rfloor$, where $p$ is the number of generators of $I$, and $q - r$ is the number of generators of $J$ not in $I$.
• For $n = 3k - 2$, the coefficients $\alpha_0, \ldots, \alpha_k$ are non-negative for small $k$, suggesting a potential exact formula, though this remains heuristic without a full decomposition.