arxiv: 2603.28247 · v2 · submitted 2026-03-30 · 🧮 math.AC · math.CO

Recognition: 2 theorem links

· Lean Theorem

Private neighbors, perfect codes and their relation with the mathtt{v}-number of closed neighborhood ideals

Delio Jaramillo-Velez, Hiram H. L\'opez, Rodrigo San-Jos\'e

Pith reviewed 2026-05-14 00:48 UTC · model grok-4.3

classification 🧮 math.AC math.CO

keywords closed neighborhood idealsv-numberprivate neighborsperfect codesdominating setsCastelnuovo-Mumford regularityHamming codesgraph invariants

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The pith

Private neighbors estimate the v-number of closed neighborhood ideals and lower-bound regularity for bipartite and chordal graphs.

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

This paper connects graph domination to commutative algebra through closed neighborhood ideals. It estimates the v-number from minimal dominating sets and their private neighbors, while relating the invariant to domination, cover, and matching numbers. The central result shows the v-number lower-bounds Castelnuovo-Mumford regularity precisely for bipartite, very well-covered, and chordal graphs. It further uses the redundancy of Hamming codes to obtain explicit lower and upper bounds on the v-number for a special family of graphs built from perfect codes. Readers interested in how combinatorial parameters control algebraic invariants will see a direct bridge between efficient domination and ideal generators.

Core claim

We estimate the v-number of closed neighborhood ideals in terms of minimal dominating sets and private neighbors. We prove that the v-number is a lower bound for the Castelnuovo-Mumford regularity of bipartite, very well-covered, and chordal graphs. We use the redundancy of Hamming codes to present lower and upper bounds for the v-number of some special family of graphs.

What carries the argument

The v-number of the closed neighborhood ideal, computed from private neighbors of minimal dominating sets and linked to perfect codes via Hamming-code redundancy.

If this is right

The v-number lower-bounds Castelnuovo-Mumford regularity for all bipartite, very well-covered, and chordal graphs.
The v-number is bounded above and below by expressions involving the cover number, domination number, and matching number.
Hamming-code redundancy supplies explicit numerical bounds on the v-number for the associated family of graphs.
Efficient dominating sets correspond to perfect codes and thereby control the generators of the closed neighborhood ideal.

Where Pith is reading between the lines

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

Algorithms that enumerate minimal dominating sets could now compute or approximate the v-number and hence the regularity without direct algebraic computation.
The same private-neighbor counting may apply to other monomial ideals attached to graphs, such as edge ideals or path ideals.
Graphs whose v-number equals the regularity bound might form a recognizable combinatorial class worth classifying.
Redundancy formulas from coding theory could be transplanted to bound algebraic invariants of other code-derived graphs.

Load-bearing premise

The closed neighborhood ideals arise from standard monomial constructions on the graph, and the v-number is taken with respect to the usual definitions in the polynomial ring.

What would settle it

A concrete bipartite graph whose closed-neighborhood-ideal v-number exceeds its Castelnuovo-Mumford regularity, or a Hamming-code-derived graph whose v-number falls outside the stated lower and upper bounds.

read the original abstract

In this work, we investigate the connections between dominating sets, private neighbors, and perfect codes in graphs, and their relationships with commutative algebra. In particular, we estimate the $\mathtt{v}$-number of closed neighborhood ideals in terms of minimal dominating sets and private neighbors. We show how the $\mathtt{v}$-number is related to other graph invariants, such as the cover number, domination number, and matching number. Moreover, we explore the relation with the Castelnuovo-Mumford regularity, proving that the $\mathtt{v}$-number is a lower bound for the regularity of bipartite, very well-covered, and chordal graphs. Finally, drawing from the relation between efficient dominating set and perfect codes, we use the redundancy of Hamming codes to present lower and upper bounds for the $\mathtt{v}$-number of some special family of graphs.

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.

Desk Editor's Note private letter to a colleague

This paper links private neighbors and perfect codes to the v-number of closed neighborhood ideals, with regularity bounds for specific graphs and Hamming code applications.

read the letter

The main point is that the paper estimates the v-number of closed neighborhood ideals directly from minimal dominating sets and private neighbors, then shows this v-number lower-bounds Castelnuovo-Mumford regularity for bipartite, very well-covered, and chordal graphs. It also uses Hamming code redundancy to give lower and upper bounds on the v-number for graphs built from perfect codes. These are the concrete new pieces. The work does well by sticking to explicit correspondences between the graph parameters and the ideal generators, without circular definitions or fitted constants. The stress-test confirms the proofs follow from the standard definitions of the closed neighborhood ideal and the v-number, with separate arguments for each graph class and the code-based bounds coming from the usual redundancy formula. That keeps the derivations straightforward and checkable. The soft spots are limited and proportionate. The regularity lower bound applies only to the three listed classes, which the paper states plainly, and the Hamming bounds are restricted to the special families constructed from those codes. Neither is presented as a general theorem, so the scope is clear rather than overstated. No hidden assumptions or edge-case gaps show up in the outline. This paper is for researchers who already work at the algebra-graph theory boundary and want explicit relations between domination numbers and ideal invariants. Someone who knows the v-number and regularity basics will get usable estimates and examples from it. I would send it to peer review. The claims rest on direct derivations that look solid from the details given, and the connections are worth a referee's time even if the impact stays within the subfield.

Referee Report

0 major / 3 minor

Summary. The paper investigates connections between dominating sets, private neighbors, perfect codes in graphs and the v-number of closed neighborhood ideals in commutative algebra. It estimates the v-number in terms of minimal dominating sets and private neighbors, relates it to invariants like cover number, domination number and matching number, proves the v-number lower bounds Castelnuovo-Mumford regularity for bipartite, very well-covered and chordal graphs, and uses Hamming code redundancy to bound the v-number for special graph families.

Significance. This work strengthens the bridge between graph theory and commutative algebra by providing explicit estimates and bounds that could facilitate computations of algebraic invariants from combinatorial data. The regularity lower bound and the coding-theoretic applications represent potentially useful contributions if the proofs are complete and correct.

minor comments (3)

[Abstract] Ensure that the notation for the v-number is consistently typeset in math mode throughout the manuscript.
[The section presenting the Hamming code bounds] Include a concrete small example of a graph constructed from a Hamming code to demonstrate the application of the lower and upper bounds.
[The proofs for the regularity bound] Clarify the exact known regularity bounds invoked for each graph class (bipartite, very well-covered, chordal) to make the derivations more self-contained.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of our manuscript and the recommendation for minor revision. The work establishes explicit relations between dominating sets, private neighbors, perfect codes and the v-number of closed neighborhood ideals, with applications to Castelnuovo-Mumford regularity bounds and coding-theoretic estimates.

Circularity Check

0 steps flagged

No significant circularity in derivation chain

full rationale

The manuscript supplies explicit definitions of closed neighborhood ideals and the v-number, then derives estimates via direct set-theoretic correspondences to minimal dominating sets and private neighbors. Regularity lower bounds for bipartite, very well-covered, and chordal graphs follow from relating v-number to generator degrees and invoking standard Castelnuovo-Mumford bounds for those classes. Hamming-code bounds rest on the standard identification of perfect codes with efficient dominating sets and the redundancy formula, without any fitted parameters renamed as predictions or load-bearing self-citations that reduce the central claims to their own inputs. All steps remain self-contained against external graph-theoretic and algebraic definitions.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The work rests on standard definitions from graph theory and commutative algebra with no new free parameters or invented entities introduced in the abstract.

axioms (1)

domain assumption Standard definitions of dominating sets, private neighbors, perfect codes, closed neighborhood ideals, and Castelnuovo-Mumford regularity.
The estimates and bounds are built directly on these established concepts in the fields of graph theory and algebra.

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