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arxiv: 2604.23428 · v2 · submitted 2026-04-25 · 🧮 math.AC · math.CO

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Associated primes of powers of closed neighborhood ideals and diameters of graphs

Ha Thi Thu Hien, Thanh Vu

Pith reviewed 2026-05-08 06:47 UTC · model grok-4.3

classification 🧮 math.AC math.CO
keywords associated primesclosed neighborhood idealgraph diametermonomial ideal powerscommutative algebraalgebraic graph theory
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The pith

If the maximal homogeneous ideal is an associated prime of the tth power of the closed neighborhood ideal of G, then the diameter of G is at most 7t-8.

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

This paper establishes a connection between the algebraic structure of powers of the closed neighborhood ideal in a graph and the graph's diameter. It shows that when the maximal ideal in the polynomial ring becomes an associated prime of the t-th power of this ideal for t at least 2, the longest shortest path between any two vertices in the graph is limited to 7t minus 8. The authors also demonstrate that this upper limit is attained by certain graphs for each such t, making the bound optimal. This matters for readers interested in how properties of monomial ideals can impose restrictions on combinatorial features of graphs.

Core claim

If the maximal homogeneous ideal is an associated prime of the tth power of the closed neighborhood ideal of G, then the diameter of G is at most 7t - 8. This bound is sharp for all t ≥ 2, as there exist graphs where the diameter equals 7t-8 while satisfying the associated prime condition.

What carries the argument

The closed neighborhood ideal of G, a monomial ideal whose t-th powers have associated primes that detect the presence of paths of length related to the exponent t in the graph.

If this is right

  • Any graph G where the maximal homogeneous ideal is an associated prime of the t-th power of the closed neighborhood ideal must have diameter at most 7t-8.
  • For every t ≥ 2 there exist graphs achieving diameter exactly 7t-8 for which the maximal ideal is an associated prime of the ideal power.
  • Algebraic tools for computing associated primes of monomial ideal powers can be used to obtain diameter information in the corresponding graphs.
  • The allowed diameter under the algebraic condition grows linearly with the exponent t.

Where Pith is reading between the lines

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

  • This algebraic certificate for diameter bounds could be implemented in computer algebra systems to check or constrain diameters in explicit graphs.
  • The factor of 7 likely reflects repeated applications of neighborhood relations or path concatenations in the proof, opening the possibility of tighter constants for restricted graph families.
  • Similar diameter controls might hold when other primes, not necessarily maximal, appear in the associated primes of the ideal powers.

Load-bearing premise

The graph is simple and connected, and the associated prime condition corresponds directly to the existence of paths of controlled length in the graph via the monomial generators.

What would settle it

A simple connected graph G with diameter larger than 7t-8 for some t ≥ 2 such that the maximal homogeneous ideal is still an associated prime of the t-th power of its closed neighborhood ideal.

Figures

Figures reproduced from arXiv: 2604.23428 by Ha Thi Thu Hien, Thanh Vu.

Figure 1
Figure 1. Figure 1: A graph achieving the maximal diameter bound which is the maximum possible distance. Therefore, diam(Gt) = 7t − 1. Let fi = xi,2 · · · xi,11 and f = f1 · · · ft . Note that NGt [supp(fi)] ∩ NGt [supp(fj )] = ∅ for all i ̸= j. Since fi ∈ NI(Gt) but fi ∈/ NI(Gt) 2 , we deduce that f ∈ NI(Gt) t but f /∈ NI(Gt) t+1. It suffices to show that xi,j ∈ NI(Gt) t+1 : f for all i = 1, . . . , t and j = 1, . . . , 12. … view at source ↗
read the original abstract

Let $G$ be a simple connected graph and $t \ge 2$ an integer. We prove that if the maximal homogeneous ideal is an associated prime of the $t$th power of the closed neighborhood ideal of $G$, then the diameter of $G$ is at most $7t - 8$. We further show that this bound is sharp for all $t \ge 2$.

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

0 major / 1 minor

Summary. The manuscript proves that if the maximal homogeneous ideal is an associated prime of the t-th power of the closed neighborhood ideal of a simple connected graph G (t ≥ 2 integer), then the diameter of G is at most 7t − 8. It further establishes sharpness of the bound for every t ≥ 2 via explicit graph constructions.

Significance. The result supplies a direct algebraic criterion, phrased in terms of associated primes of monomial ideal powers, that forces an upper bound on graph diameter. The argument proceeds from the standard definition of associated primes via colon ideals, translated into controlled path lengths between vertices, and the explicit sharpness examples confirm that the constant 7t − 8 cannot be improved in general. This supplies a clean, falsifiable link between commutative-algebraic invariants and a basic graph-theoretic quantity.

minor comments (1)
  1. The notation for the closed neighborhood ideal I(G) and the maximal ideal m is introduced cleanly, but a single sentence recalling the monomial generators of I(G) in the first paragraph of the introduction would aid readers whose primary background is graph theory.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the manuscript, the clear summary of its main results, and the recommendation to accept. No major comments were raised.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The derivation proceeds directly from the standard definition of associated primes of monomial ideals (via colon ideals (I^t : f) = m) and the correspondence between monomial generators and closed neighborhoods in G. The diameter bound 7t-8 is obtained by an explicit length estimate on paths required to absorb the exponents of a witness monomial f, using only the connectedness of G and the definition of closed-neighborhood generators; no quantity is defined in terms of the target diameter, no parameters are fitted, and no self-citation chain or imported uniqueness theorem is invoked. Sharpness follows from an explicit family of graphs attaining equality. All steps remain within the given hypotheses and use only ordinary facts about monomial primary decomposition.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper rests on the standard definition of associated primes in graded polynomial rings and the combinatorial definition of closed neighborhoods and diameter; no free parameters are fitted and no new entities are postulated.

axioms (2)
  • standard math Standard properties of associated primes for monomial ideals in polynomial rings over fields
    Invoked to identify when the maximal homogeneous ideal is associated to I(G)^t.
  • domain assumption G is a simple connected undirected graph
    Required for the closed neighborhood ideal to be well-defined and for diameter to be finite.

pith-pipeline@v0.9.0 · 5350 in / 1272 out tokens · 54541 ms · 2026-05-08T06:47:07.294704+00:00 · methodology

discussion (0)

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Reference graph

Works this paper leans on

14 extracted references · 1 canonical work pages

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