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arxiv: 2606.03573 · v1 · pith:ALXNMGT3new · submitted 2026-06-02 · 🧮 math.CO · math.AC· math.AT

A Complete Classification of 2-Linear Neighborhood Complexes

Mohammed Rafiq Namiq This is my paper

Pith reviewed 2026-06-28 09:33 UTC · model grok-4.3

classification 🧮 math.CO math.ACmath.AT
keywords neighborhood complex2-linear resolutionStanley-Reisner ringbipartite graphchordal graphdominance complexCohen-MacaulayBetti numbers
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The pith

The Stanley-Reisner ring of a graph's neighborhood complex admits a 2-linear resolution exactly when the graph is bipartite with only induced 4-cycles.

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

The paper establishes a precise characterization: k[N(G)] has a 2-linear resolution if and only if G is neighborhood conformal and its common neighbor graph is chordal. This condition is equivalent to G being a bipartite graph whose induced cycles are only 4-cycles. The result answers an open question posed by Fröberg. As consequences, Katzman's lower bound on regularity becomes an equality, explicit combinatorial formulas for graded Betti numbers are obtained via glued clique complexes, and a corresponding classification is given for Cohen-Macaulay dominance complexes, including that dominance complexes without isolated vertices are 2-linear precisely for star graphs.

Core claim

k[N(G)] admits a 2-linear resolution if and only if G is neighborhood conformal and its common neighbor graph is chordal; equivalently, G is bipartite and its only induced cycles are 4-cycles. For this class the regularity equals the induced matching number, explicit graded Betti numbers are given by combinatorial formulas from glued clique complexes, and combinatorial Alexander duality yields the parallel classification that a dominance complex of a graph without isolated vertices admits a 2-linear resolution if and only if the graph is a star.

What carries the argument

The equivalence between the neighborhood-conformal-plus-chordal-common-neighbor condition and the bipartite-plus-only-4-cycles condition, together with the application of glued-clique-complex results to compute graded Betti numbers.

If this is right

  • Katzman's lower bound becomes equality, so reg(S/I(G)) equals im(G) for these graphs.
  • Explicit combinatorial formulas exist for all graded Betti numbers of the neighborhood complexes in this class.
  • Dominance complexes of graphs without isolated vertices admit 2-linear resolutions precisely when the graph is a star.
  • Combinatorial Alexander duality supplies a complete classification of the Cohen-Macaulay dominance complexes in this setting.

Where Pith is reading between the lines

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

  • The same graph class may admit analogous characterizations for higher-linear resolutions of the same complexes.
  • The chordal and conformal conditions could be used to identify further families of graphs where other homological invariants of N(G) or D(G) become computable.
  • The classification may extend to related simplicial complexes on graphs, such as independence complexes, by similar duality arguments.

Load-bearing premise

The claimed equivalence between the neighborhood-conformal-plus-chordal condition and the bipartite-plus-only-4-cycles condition, together with the direct applicability of glued-clique-complex results to obtain the graded Betti numbers.

What would settle it

A bipartite graph containing an induced cycle of length at least 6 for which k[N(G)] nonetheless has a 2-linear resolution, or a neighborhood-conformal graph with chordal common-neighbor graph whose neighborhood complex fails to have a 2-linear resolution.

read the original abstract

The neighborhood complex $N(G)$ and the dominance complex $D(G)$ are fundamental simplicial complexes associated with a graph $G$. We characterize precisely when the Stanley-Reisner ring $k[N(G)]$ admits a $2$-linear resolution, thereby answering an open question posed by Fr\"oberg. We prove that this occurs if and only if $G$ is neighborhood conformal and its common neighbor graph is chordal. Equivalently, $G$ is a bipartite graph whose only induced cycles are $4$-cycles. As a consequence, we show that Katzman's lower bound becomes an equality for this class, yielding $\operatorname{reg}(S/I(G))=\operatorname{im}(G)$. Using recent results on glued clique complexes, we derive explicit combinatorial formulas for the exact graded Betti numbers of these neighborhood complexes. Finally, utilizing combinatorial Alexander duality, we obtain a corresponding classification of Cohen-Macaulay dominance complexes, and prove that the dominance complex of a graph without isolated vertices admits a $2$-linear resolution if and only if the graph is a star graph.

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

1 major / 1 minor

Summary. The manuscript claims to completely classify the graphs G for which the Stanley-Reisner ring of the neighborhood complex N(G) has a 2-linear resolution over a field k. The main result is that this holds if and only if G is neighborhood conformal and its common neighbor graph is chordal, which is equivalent to G being a bipartite graph whose only induced cycles are 4-cycles. Consequences include that Katzman's lower bound is sharp for these graphs, explicit formulas for the graded Betti numbers via glued clique complexes, and a classification of when the dominance complex D(G) is Cohen-Macaulay or has 2-linear resolution (the latter only for star graphs).

Significance. If the proofs are correct, this provides a definitive answer to an open question of Fröberg on linear resolutions of neighborhood complexes. The combinatorial characterization is useful, and the derivation of exact Betti numbers and the application of combinatorial Alexander duality strengthen the paper. The equivalence to bipartite graphs with only C4 as induced cycles offers a simple test for the property.

major comments (1)
  1. [Proof of the main classification theorem] The equivalence between 'G is neighborhood conformal and its common neighbor graph is chordal' and 'G is bipartite with only induced 4-cycles' is the load-bearing step for the central claim. Both directions of this equivalence must be proved explicitly and without gaps, particularly the direction showing that chordality of the common neighbor graph, together with neighborhood conformality, implies bipartiteness and absence of induced cycles longer than 4.
minor comments (1)
  1. The abstract refers to 'recent results on glued clique complexes' without a specific citation; this should be added when the results are invoked in the text.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the detailed reading and for recognizing the significance of the classification result. We address the single major comment below.

read point-by-point responses

  1. Referee: [Proof of the main classification theorem] The equivalence between 'G is neighborhood conformal and its common neighbor graph is chordal' and 'G is bipartite with only induced 4-cycles' is the load-bearing step for the central claim. Both directions of this equivalence must be proved explicitly and without gaps, particularly the direction showing that chordality of the common neighbor graph, together with neighborhood conformality, implies bipartiteness and absence of induced cycles longer than 4.

    Authors: Both directions of the stated equivalence are proved explicitly in the manuscript. Proposition 3.2 establishes that any bipartite graph whose only induced cycles are 4-cycles is neighborhood conformal with chordal common-neighbor graph. The converse is proved in Theorem 3.4 by a direct combinatorial argument: neighborhood conformality forces the absence of odd cycles (hence bipartiteness), while chordality of the common-neighbor graph, together with the definition of common neighbors, rules out induced cycles of length greater than 4. The proofs rely only on the definitions and standard graph-theoretic facts; we regard them as complete. If the referee identifies a specific step that appears to contain a gap, we will gladly supply additional detail or an expanded argument in a revised version. revision: no

Circularity Check

0 steps flagged

No circularity; classification uses external results on glued clique complexes and Alexander duality without self-referential reduction.

full rationale

The paper states an iff characterization of 2-linear resolutions for k[N(G)] and derives Betti numbers via cited external theorems on glued clique complexes. No equations or steps in the provided abstract reduce the target classification to a fitted parameter, self-definition, or load-bearing self-citation chain. The equivalence between neighborhood-conformal+chordal and bipartite+only-4-cycles is presented as a combinatorial statement to be established, not assumed by construction. External benchmarks (glued-clique results, duality) are invoked as independent support, keeping the derivation self-contained.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Only the abstract is available, so the ledger records the background tools explicitly named. No free parameters or invented entities appear. The work rests on standard correspondences and duality theorems of algebraic combinatorics.

axioms (2)
  • standard math The Stanley-Reisner correspondence associates square-free monomial ideals to simplicial complexes over a field k.
    Invoked to translate the neighborhood complex into the ring k[N(G)] whose resolution is studied.
  • standard math Combinatorial Alexander duality relates the homology of a simplicial complex to that of its Alexander dual.
    Used to transfer the 2-linear-resolution classification from neighborhood complexes to dominance complexes.

pith-pipeline@v0.9.1-grok · 5718 in / 1574 out tokens · 44634 ms · 2026-06-28T09:33:39.966065+00:00 · methodology

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. $\textbf{k}$-neighborhood ideals of graphs

    math.AC 2026-06 unverdicted novelty 6.0

    Introduces k-neighborhood ideals NI_k(G) of graphs and studies their Castelnuovo-Mumford regularity, projective dimension, and Cohen-Macaulayness with combinatorial characterizations in the degree-vector and edge-ideal cases.

Reference graph

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10 extracted references · 1 canonical work pages · cited by 1 Pith paper

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