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arxiv: 2606.08159 · v1 · pith:RRT2LZXNnew · submitted 2026-06-06 · 🧮 math.AC · math.CO

k-neighborhood ideals of graphs

Somayeh Moradi , Leila Sharifan This is my paper

Pith reviewed 2026-06-27 18:54 UTC · model grok-4.3

classification 🧮 math.AC math.CO
keywords k-neighborhood idealmonomial idealCastelnuovo-Mumford regularityprojective dimensionCohen-Macaulay propertyedge idealgraph ideal
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The pith

The k-neighborhood ideal of a graph yields combinatorial characterizations of its regularity and projective dimension when k is the degree vector or the ideal equals an edge ideal.

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

This paper defines the k-neighborhood ideal NI_k(G) as the squarefree monomial ideal obtained by summing, over each vertex i, all products of exactly k_i distinct variables from the closed neighborhood of i. The authors study its Castelnuovo-Mumford regularity, projective dimension, and Cohen-Macaulay property. Special attention is given to the case k equal to the degree vector of G and to the case in which NI_k(G) coincides with an edge ideal; in these settings the paper supplies combinatorial characterizations and bounds on the invariants for several classes of graphs and examines when the ideals are Cohen-Macaulay.

Core claim

We introduce the k-neighborhood ideal NI_k(G) = sum over i of the ideal generated by all squarefree monomials x_W of degree k_i with W contained in the closed neighborhood of vertex i, under the standing hypothesis 1 ≤ k_i ≤ deg(i) + 1. For the degree-vector choice of k and for the choice that makes NI_k(G) an edge ideal we provide combinatorial characterizations and bounds for the regularity and projective dimension of NI_k(G) on several classes of graphs and we investigate the Cohen-Macaulay property of these ideals.

What carries the argument

The k-neighborhood ideal NI_k(G), the squarefree monomial ideal assembled by taking all degree-k_i products inside each closed neighborhood N_G[i].

If this is right

  • For the degree-vector case the regularity of NI_k(G) is bounded by a combinatorial function of the graph for the classes considered.
  • The projective dimension of NI_k(G) likewise admits a combinatorial bound or exact value in the degree-vector and edge-ideal cases.
  • The Cohen-Macaulay property of NI_k(G) can be decided by graph-theoretic conditions in the settings studied.
  • When NI_k(G) is an edge ideal the same homological invariants are controlled by the underlying graph structure.

Where Pith is reading between the lines

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

  • The construction supplies a uniform algebraic framework that includes both closed-neighborhood ideals and edge ideals, so other choices of the vector k may produce further families whose invariants are likewise graph-computable.
  • The same neighborhood-based generators could be used to relate regularity to classical graph parameters such as domination number or matching number on additional graph families.
  • Because the definition is local to each closed neighborhood, the method may extend directly to the study of induced subgraphs or to the comparison of NI_k(G) with its restriction to an induced subgraph.

Load-bearing premise

The condition 1 ≤ k_i ≤ deg_G(i) + 1 for each vertex i must hold so that the generators x_W are well-defined squarefree monomials of the stated degree.

What would settle it

An explicit minimal free resolution or direct regularity computation for a concrete graph (for example a path or a tree) with k equal to the degree vector that produces a regularity value different from the combinatorial expression given in the paper.

Figures

Figures reproduced from arXiv: 2606.08159 by Leila Sharifan, Somayeh Moradi.

Figure 1
Figure 1. Figure 1: A clique-star graph with the central clique K6. A graph G is called a generalized caterpillar graph if there exists a path graph P such that G is obtained by attaching complete graphs K1, K2, . . . , Kt to P, where each attachment is performed by identifying either a vertex or an edge of Ki with a vertex or an edge of P. An example of a generalized caterpillar graph is illustrated in [PITH_FULL_IMAGE:figu… view at source ↗
Figure 2
Figure 2. Figure 2: A generalized caterpillar graph obtained from a path graph P7. Theorem 3.6. Let G be one of the following graphs: 10 [PITH_FULL_IMAGE:figures/full_fig_p010_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: A chordal graph. Proposition 4.6. If G is a block graph, then Ge is a chordal graph. Proof. Let G be a block graph. We prove the assertion by induction on n = |V (G)|. If n = 1, there is nothing to prove. Let n > 1 and consider the clique complex ∆(G) of G. Since G is chordal, by [10, Lemma 3.1], ∆(G) has a leaf, say F. So there exists a maximal clique F ′ of G such that F ∩ F ′′ ⊆ F ∩ F ′ for any maximal … view at source ↗
read the original abstract

In this paper, we introduce and investigate the $\textbf{k}$-neighborhood ideal of a graph, a natural generalization of the closed neighborhood ideal. Let $G$ be a simple graph on the vertex set $[n]$, and let $S=K[x_1,\dots,x_n]$ be the polynomial ring over a field $K$. For a vector $\textbf{k}=(k_1,\ldots,k_n)\in \mathbb{N}^n$ satisfying $1\leq k_i\leq \textrm{deg}_G(i)+1$ for all $i$, the $\textbf{k}$-neighborhood ideal of $G$ is defined as the squarefree monomial ideal $$\textrm{NI}_{\textbf{k}}(G)=\sum_{i=1}^n\, (\textbf{x}_W:\, W\subseteq N_G[i],\, |W|=k_i)$$ of $S$, where $\textbf{x}_W=\prod_{i\in W} x_i$. We study homological invariants and properties of $\textrm{NI}_{\textbf{k}}(G)$ focusing on its Castelnuovo-Mumford regularity, projective dimension and Cohen-Macaulayness. Special attention is devoted to the case where the vector ${\textbf{k}}$ is the degree-vector of the graph, i.e., $k_i=\textrm{deg}_G(i)$ for all vertices $i$, and to the case where $\textrm{NI}_{\textbf{k}}(G)$ coincides with the edge ideal of a graph. In these settings, we provide combinatorial characterizations and bounds for the regularity and projective dimension of $\textrm{NI}_{\textbf{k}}(G)$ for several classes of graphs, and further investigate the Cohen-Macaulay property of these ideals.

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 / 2 minor

Summary. The paper introduces the k-neighborhood ideal NI_k(G) of a graph G as the squarefree monomial ideal generated by summing, for each vertex i, the ideal generated by all products of exactly k_i variables from the closed neighborhood N_G[i]. It focuses on the Castelnuovo-Mumford regularity, projective dimension, and Cohen-Macaulay property of these ideals, providing combinatorial characterizations and bounds for the case when k is the degree vector of G and when NI_k(G) is an edge ideal, for several classes of graphs.

Significance. If the results hold, this generalization of neighborhood ideals offers combinatorial methods to bound and characterize important homological invariants for monomial ideals arising from graphs. This could be significant for researchers working on edge ideals and their generalizations in algebraic combinatorics, as it extends known results on regularity and projective dimension.

minor comments (2)
  1. [Abstract] The abstract claims combinatorial characterizations for several classes of graphs but does not specify which classes; listing them would improve clarity.
  2. The condition 1 ≤ k_i ≤ deg_G(i)+1 is well-motivated for ensuring squarefree generators, but a brief remark on whether the homological results extend beyond this range (or why they do not) would aid readers.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary of our work and the recommendation of minor revision. The report lists no major comments, so we have no specific points requiring point-by-point response or revision.

Circularity Check

0 steps flagged

No significant circularity; definition and combinatorial results are independent

full rationale

The paper introduces the k-neighborhood ideal via an explicit sum over subsets of closed neighborhoods, with the 1 ≤ k_i ≤ deg(i)+1 condition serving only to ensure generators are squarefree monomials of the stated degree. Subsequent claims consist of combinatorial characterizations and bounds on reg(NI_k(G)) and pd(NI_k(G)) plus Cohen-Macaulayness for the degree-vector case and edge-ideal case; these rest on direct graph-theoretic arguments rather than any reduction to fitted parameters, self-citations, or prior ansatzes from the same authors. No equations or steps in the provided text equate a derived invariant to an input by construction. The derivation chain is therefore self-contained against external combinatorial data.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

Only the abstract is available, so the ledger records the minimal background needed for the stated definition; no numerical fitting occurs.

axioms (1)
  • standard math Polynomial ring S = K[x1,...,xn] over a field K admits squarefree monomial ideals generated by products of distinct variables.
    The construction of NI_k(G) as a sum of such monomials relies on this standard setup.
invented entities (1)
  • k-neighborhood ideal NI_k(G) no independent evidence
    purpose: Generalize the closed neighborhood ideal and serve as the object whose homological invariants are studied.
    Newly defined via the sum over neighborhood subsets of size k_i.

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Reference graph

Works this paper leans on

32 extracted references · 9 canonical work pages · 2 internal anchors

  1. [1]

    C. N. Campos, Y. Wakabayashi, On dominating sets of maximal outerplanar graphs,Discrete Appl. Math.161 (2013), no. 3, pp. 330-335

    2013

  2. [2]

    Caviglia, H.T

    G. Caviglia, H.T. H` a, J. Herzog, M. Kummini, N. Terai, N.V. Trung, Depth and regularity modulo a principal ideal, J. Algebraic Combin. 49 (2019), no. 1, 1–20

    2019

  3. [3]

    Chakraborty, A.P

    S. Chakraborty, A.P. Joseph, A. Roy, A. Singh, Castelnuovo-Mumford regularity of the closed neighborhood ideal of a graph, J. Algebraic Combin. 61 (2025), no. 1, Paper No. 1, 18 pp

    2025

  4. [4]

    Fr¨ oberg, Stanley–Reisner rings of neighborhood complexes and linear resolutions, J

    R. Fr¨ oberg, Stanley–Reisner rings of neighborhood complexes and linear resolutions, J. Alge- braic Combin. (2026) doi.org/10.1007/s10801-026-01530-x

    work page doi:10.1007/s10801-026-01530-x 2026

  5. [5]

    H.T. H` a, A. Van Tuyl, Monomial ideals, edge ideals of hypergraphs, and their graded Betti numbers, J. Algebraic Combin. 27 (2008), no. 2, 215–245

    2008

  6. [6]

    Harary, T

    F. Harary, T. W. Haynes, Double domination in graphs, Ars Combin. 55 (2000), 201–213

    2000

  7. [7]

    H.T.T. Hien, T. Vu, Associated primes of the second power of closed neighborhood ideals of graphs, J. Algebra Appl. (2026), 2750187

    2026

  8. [8]

    Herzog, A generalization of the Taylor complex construction, Comm

    J. Herzog, A generalization of the Taylor complex construction, Comm. Algebra 35 (2007), no. 5, 1747–1756

    2007

  9. [9]

    Herzog, T

    J. Herzog, T. Hibi,Monomial ideals, Graduate texts in Mathematics 260, Springer, 2011

    2011

  10. [10]

    Herzog, T

    J. Herzog, T. Hibi, X. Zheng, Dirac’s theorem on chordal graphs and Alexander duality, European J. Combin. 25 (2004), 949-960

    2004

  11. [11]

    Herzog, T

    J. Herzog, T. Hibi, X. Zheng, Cohen-Macaulay chordal graphs, J. Combin. Theory Ser. A 113 (2006), 911–916

    2006

  12. [12]

    Honeycutt, K

    J. Honeycutt, K. Sather-Wagstaff, Closed neighborhood ideals of finite simple graphs, Matem- atica 1(2) (2022), 387–394

    2022

  13. [13]

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

    D. Jaramillo-Velez, H. H. L´ opez, R. San-Jos´ e, Private neighbors, perfect codes and their relation with the v-number of closed neighborhood ideals, arXiv:2603.28247 (2026). 23

    work page internal anchor Pith review Pith/arXiv arXiv 2026

  14. [14]

    Joseph, A

    A.P. Joseph, A. Roy, A. Singh, Barile-Macchia Resolutions and the closed neighborhood ideal, arXiv:2511.22657 (2025)

    work page arXiv 2025

  15. [15]

    Leaman, Cohen-Macaulay properties of closed neighborhood ideals, M.S

    J. Leaman, Cohen-Macaulay properties of closed neighborhood ideals, M.S. Thesis, Clemson University (2023)

    2023

  16. [16]

    Lim, Geometrically vertex decomposable open neighborhood ideals, arXiv:2512.12886 (2025)

    J. Lim, Geometrically vertex decomposable open neighborhood ideals, arXiv:2512.12886 (2025)

    work page arXiv 2025

  17. [17]

    J. Lim, J. Gossell, K. Sather-Wagstaff, D. Adams, V. Nguyen, S. Castro-Tarabulsi, A. Herbert, Y. Qtan, M. Schaller, Z. Zhou, Y. Zhuo, Open neighborhood ideals of well totally dominated trees are Cohen-Macaulay, arXiv:2510.19707 (2025)

    work page arXiv 2025

  18. [18]

    Matsushita, S

    T. Matsushita, S. Wakatsuki, Dominance complexes, neighborhood complexes and combina- torial alexander duals, J. Combin.Theory, Ser. A 211 (2025), 105978

    2025

  19. [19]

    Moradi, L

    S. Moradi, L. Sharifan, On homological invariants and Cohen-Macaulayness of closed neigh- borhood ideals, arXiv:2602.07910 (2026)

    work page arXiv 2026

  20. [20]

    M. A. Nambi, A. A. Qureshi, Squarefree powers of closed neighborhood ideals, arXiv preprint arXiv:2603.15229 (2026)

    work page arXiv 2026

  21. [21]

    M. R. Namiq, A complete classification of 2-linear neighborhood complexes, arXiv preprint arXiv:2606.03573 (2026)

    work page internal anchor Pith review Pith/arXiv arXiv 2026

  22. [22]

    Nasernejad, S

    M. Nasernejad, S. Bandari, L. G. Roberts, Normality and associated primes of closed neigh- borhood ideals and dominating ideals, J. Algebra Appl. 24, no. 1 (2025)

    2025

  23. [23]

    Nasernejad, A.A

    M. Nasernejad, A.A. Qureshi, Algebraic implications of neighborhood hypergraphs and their transversal hypergraphs, Comm. Algebra 6 (2024), 2328-2345

    2024

  24. [24]

    Nasernejad, A.A.Qureshi, S

    M. Nasernejad, A.A.Qureshi, S. Bandari, A. Musapa¸ sao˘ glu, Dominating ideals and closed neighborhood ideals of graphs, Mediterr. J. Math. 19 (2022), no. 4, 152

    2022

  25. [25]

    Olteanu, O

    A. Olteanu, O. Olteanu, On the closed neighborhood ideal of the square of the path graph, arXiv:2602.14163 (2026)

    work page arXiv 2026

  26. [26]

    Sharifan, Minimal free resolution of monomial ideals by iterated mapping cone, Bull

    L. Sharifan, Minimal free resolution of monomial ideals by iterated mapping cone, Bull. Iranian Math. Soc. 44 (2018), no. 4, 1007–1024

    2018

  27. [27]

    Sharifan,t-closed neighborhood ideal of a graph, Beitr

    L. Sharifan,t-closed neighborhood ideal of a graph, Beitr. Algebra Geom. 66 (2025), 927–940

    2025

  28. [28]

    Sharifan, S

    L. Sharifan, S. Moradi, Closed neighborhood ideal of a graph, Rocky Mountain J. Math. 50 (2020), no. 3, 1097–1107

    2020

  29. [29]

    Sharifan, M

    L. Sharifan, M. Varbaro, Graded Betti numbers of ideals with linear quotients, Matematiche (Catania) 63 (2008), no. 2, 257–265

    2008

  30. [30]

    Terai, Alexander duality theorem and Stanley-Reisner rings

    N. Terai, Alexander duality theorem and Stanley-Reisner rings. Free resolutions of coordinate rings of projective varieties and related topics (Japanese) (Kyoto, 1998). S¨ urikaisekikenky¨ usho K¨ oky¨ uroku no. 1078 (1999), 174–184

    1998

  31. [31]

    J. Topp, L. Volkmann, On domination and independence numbers of graphs, Results in Math- ematics 17 (1990), 333–341

    1990

  32. [32]

    Zheng, Resolutions of facet ideals, Comm

    X. Zheng, Resolutions of facet ideals, Comm. Algebra 32 (2004), 2301-2324. Somayeh Moradi, Department of Mathematics, F aculty of Science, Ilam Univer- sity, P.O.Box 69315-516, Ilam, Iran Email address:so.moradi@ilam.ac.ir Leila Sharifan, Department of Mathematics and Computer Sciences, Hakim Sabze- vari University, P.O. Box 397, Sabzevar, Iran Email addr...

    2004