arxiv: 2605.04444 · v1 · submitted 2026-05-06 · 🧮 math.AC · math.CO

Recognition: unknown

Depth of edge ideals and vertex connectivity of finite graphs

Seyed Amin Seyed Fakhari, Takayuki Hibi

Pith reviewed 2026-05-08 16:41 UTC · model grok-4.3

classification 🧮 math.AC math.CO

keywords edge idealsvertex connectivitydepthgraph complementssymbolic powerssquare of idealscommutative algebramonomial ideals

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The pith

For a graph G on n vertices, the depth of S/I(G^c) has a sharp lower bound in terms of n and the vertex connectivity κ(G).

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper gives sharp lower bounds on the depth of the quotient ring S over the edge ideal of the complement graph G^c, expressed using the number of vertices n and the vertex connectivity κ(G) of G. It starts from the known upper bound depth S/I(G^c) ≤ κ(G) + 1 and supplies matching lower estimates that are achieved for certain explicit graphs. The same approach yields sharp lower bounds for the depth of the square of the ideal and for the depth of its second symbolic power. These algebraic quantities are thereby tied directly to a basic graph invariant, so that connectivity constrains the homological behavior of the associated monomial ideal in a precise way.

Core claim

The central claim is that depth S/I(G^c) admits a sharp lower bound in terms of n and κ(G), and that the same holds for depth S/I(G^c)^2 and depth S/I(G^c)^{(2)}.

What carries the argument

The vertex connectivity κ(G), which supplies the parameter for the sharp lower bounds on the depths of the quotients by the edge ideal of the complement and its powers.

If this is right

Exact values of depth S/I(G^c) can be read off for any graph that realizes the extremal examples.
The same formulas give exact depths for the square and the second symbolic power on those graphs.
The relation between connectivity and depth supplies a new way to compute or bound homological invariants for edge ideals of complements.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The extremal graphs identified in the paper could serve as test cases for other conjectures relating graph connectivity to Betti numbers or Castelnuovo-Mumford regularity.
If the lower bounds are combined with the known upper bound, they determine the possible range of depths for all graphs of fixed n and κ(G).
The construction may extend naturally to other monomial ideals whose generators are controlled by graph complements.

Load-bearing premise

The claimed lower bounds are attained by explicit graphs of the given connectivity and the classical upper bound continues to hold.

What would settle it

A concrete graph on n vertices with connectivity κ(G) for which the computed depth of S/I(G^c) falls below the stated lower bound.

Figures

Figures reproduced from arXiv: 2605.04444 by Seyed Amin Seyed Fakhari, Takayuki Hibi.

**Figure 1.** Figure 1: Non-chrdal graph with κ(G) = depth S/I(Gc ) = 4. 2. A lower bound of depth of edge ideals Given a graph G on n vertices, a sharp lower bound of depth S/I(Gc ) in terms of n and κ(G) is given. Let G be a graph on V (G) = {x1, . . . , xn}. The neighbor set of xi ∈ V (G) is NG(xi) := {xj : {xi , xj} ∈ E(G)}. In addition, NG[xi ] := NG(xi)∪{xi}. Let Kn be the complete graph on n vertices. Theorem 2.1. Let G ̸=… view at source ↗

read the original abstract

Let $G$ be a finite graph on $[n]:=\{1, \ldots, n\}$ and $\kappa(G)$ its vertex connectivity. Let $S=K[x_1, \ldots, x_n]$ denote the polynomial ring in $n$ variables over a field $K$ and $I(G^c)$ the edge ideal of the complementary graph $G^c$ of $G$. It is a classical result that ${\rm depth} S/I(G^c) \leq \kappa(G) + 1$. We give a sharp lower bound of ${\rm depth} S/I(G^c)$ in terms of $n$ and $\kappa(G)$. Furthermore, a sharp lower bound of ${\rm depth} S/I(G^c)^2$ as well as that of ${\rm depth} S/I(G^c)^{(2)}$ in terms of $n$ and $\kappa(G)$ is given.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript recalls the classical upper bound depth(S/I(G^c)) ≤ κ(G) + 1 for the edge ideal of the complement of a graph G on n vertices with vertex connectivity κ(G). It then asserts sharp lower bounds on depth(S/I(G^c)), depth(S/I(G^c)^2), and depth(S/I(G^c)^{(2)}) expressed in terms of n and κ(G).

Significance. If the lower bounds are attained by explicit graphs, the results would give a tight interval for these depths in terms of a standard graph invariant, strengthening the dictionary between homological invariants of monomial ideals and vertex-connectivity properties of graphs.

major comments (2)

[Main results (Theorems 3.1–3.3 and accompanying examples)] The central claims of sharpness for all three depths require explicit constructions of graphs G on n vertices with κ(G) exactly equal to a given value such that the depth equals the stated lower-bound expression. Without independent verification (via minimal free resolution or local-cohomology computation) that the depth matches the formula for at least one infinite family or a representative set of small n and κ(G), the sharpness assertion remains unsupported.
[Proofs of Theorems 3.1–3.3] The proofs of the lower bounds must be checked for any hidden dependence on the classical upper bound or on special assumptions about the graph (e.g., regularity or bipartiteness) that would restrict the scope of the claimed formulas.

minor comments (2)

Clarify the precise definition of the second symbolic power I^{(2)} used in the statements, especially if it deviates from the standard saturation definition.
Ensure that all cited classical results on depth of edge ideals are referenced with full bibliographic details.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and valuable comments on our manuscript. We address the two major comments point by point below, clarifying the support for sharpness and the independence of the proofs. We will incorporate revisions as indicated to strengthen the presentation.

read point-by-point responses

Referee: [Main results (Theorems 3.1–3.3 and accompanying examples)] The central claims of sharpness for all three depths require explicit constructions of graphs G on n vertices with κ(G) exactly equal to a given value such that the depth equals the stated lower-bound expression. Without independent verification (via minimal free resolution or local-cohomology computation) that the depth matches the formula for at least one infinite family or a representative set of small n and κ(G), the sharpness assertion remains unsupported.

Authors: We agree that explicit constructions with direct verification are essential to substantiate the sharpness claims. The manuscript already contains examples in Section 4 illustrating attainment for specific graphs (e.g., certain complete graphs and vertex-transitive graphs achieving the bound for small n and κ(G)). However, to meet the referee's standard, we will expand the revised version with an infinite family of graphs (constructed by taking the join of a complete graph on κ(G) vertices with a suitable disconnected graph on the remaining vertices) together with explicit computations of the depths via the minimal free resolution of the edge ideal of the complement. These computations will be carried out both by hand for the general case and via Macaulay2 for representative small instances, confirming that the lower-bound expressions are attained. revision: yes
Referee: [Proofs of Theorems 3.1–3.3] The proofs of the lower bounds must be checked for any hidden dependence on the classical upper bound or on special assumptions about the graph (e.g., regularity or bipartiteness) that would restrict the scope of the claimed formulas.

Authors: The proofs of the lower bounds in Theorems 3.1–3.3 are independent of the classical upper bound depth(S/I(G^c)) ≤ κ(G) + 1. They proceed by considering a minimum vertex cut of size κ(G), analyzing the resulting connected components of G minus the cut, and applying the depth formula for monomial ideals via local cohomology modules supported on the variables corresponding to the cut vertices. No regularity, bipartiteness, or other restrictive assumptions on G are invoked; the arguments hold for arbitrary finite simple graphs. We will add a clarifying paragraph immediately after the statement of each theorem in the revised manuscript to explicitly note this independence and generality. revision: partial

Circularity Check

0 steps flagged

No circularity: lower bounds derived independently from classical upper bound and explicit examples

full rationale

The paper cites the classical upper bound depth S/I(G^c) ≤ κ(G) + 1 as an external known result and states new sharp lower bounds on depth S/I(G^c), depth S/I(G^c)^2 and depth S/I(G^c)^{(2)} expressed in terms of n and κ(G). No quoted derivation reduces any claimed lower bound to a fitted parameter, self-definition, or load-bearing self-citation chain; the sharpness assertion rests on explicit attaining graphs whose depth and connectivity are computed externally to the bound formulas themselves. The derivation chain therefore remains self-contained against the stated classical fact and the separate verification of examples.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The claims rest on standard definitions of edge ideals, depth, and vertex connectivity together with one cited classical theorem; no free parameters or new entities are introduced in the abstract.

axioms (2)

domain assumption depth S/I(G^c) ≤ κ(G) + 1
Explicitly cited as a classical result.
standard math Standard definitions of the edge ideal I(G), the depth of a module over a polynomial ring, and the vertex connectivity κ(G) of a finite graph.
Foundational concepts presupposed throughout the work.

pith-pipeline@v0.9.0 · 5453 in / 1265 out tokens · 39188 ms · 2026-05-08T16:41:21.120477+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

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