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arxiv: 2606.29753 · v1 · pith:BIIHRX2Gnew · submitted 2026-06-29 · 🧮 math.CO · math.GT

A Complete Classification of Discrete d-Pseudomanifolds with at Most 2d+7 Vertices

Biplab Basak , Debolina Ghosh , Raju Kumar Gupta This is my paper

Pith reviewed 2026-06-30 05:50 UTC · model grok-4.3

classification 🧮 math.CO math.GT
keywords discrete pseudomanifoldsflag normal pseudomanifoldssimplicial spherescombinatorial topologyvertex-minimal complexesRP2 suspensions
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The pith

Discrete d-pseudomanifolds with at most 2d+6 vertices are precisely the edge graphs of flag normal simplicial d-spheres.

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

The paper defines a discrete d-pseudomanifold recursively: the induced neighborhood of every vertex must itself be a discrete (d-1)-pseudomanifold, with the base case an n-cycle for n at least 4. It first shows that any such object requires at least 2(d+1) vertices. The authors then classify all examples with up to 2d+6 vertices by enumerating their combinatorial types and prove these graphs are exactly the 1-skeletons of flag normal d-pseudomanifolds, each of which must be a simplicial d-sphere. The classification is shown to be tight for d at least 3 by constructing examples on 2d+7 vertices that are flag triangulations of the (d-2)-fold suspension of RP squared rather than spheres.

Core claim

Every discrete d-pseudomanifold has at least 2(d+1) vertices. Those with at most 2d+6 vertices are exactly the edge graphs of flag normal d-pseudomanifolds, and each such complex is a simplicial d-sphere. For d greater than or equal to 3, every flag normal d-pseudomanifold with at most 2d+7 vertices is either a simplicial d-sphere or a flag triangulation of the (d-2)-fold suspension of RP squared.

What carries the argument

The recursive local condition that the induced neighborhood graph of each vertex is a discrete (d-1)-pseudomanifold (base case: n-cycle with n at least 4).

If this is right

  • All discrete d-pseudomanifolds on at most 2d+6 vertices arise as 1-skeletons of simplicial d-spheres.
  • There is a bijection between discrete d-pseudomanifolds and edge graphs of flag normal d-pseudomanifolds in this vertex range.
  • The sphere characterization for flag normal d-pseudomanifolds holds exactly up to 2d+6 vertices and fails at 2d+7 for d at least 3 via explicit RP2 suspension examples.

Where Pith is reading between the lines

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

  • The same recursive neighborhood condition might classify additional objects once the flag or normality assumptions are dropped.
  • The appearance of suspended RP2 triangulations at the next vertex count suggests that non-orientable examples enter the classification in a controlled way.
  • One could test whether the minimal vertex count 2(d+1) remains sharp when the base cycle length is allowed to vary or when higher-dimensional links are required to satisfy extra properties.

Load-bearing premise

Requiring only that every vertex neighborhood satisfies the lower-dimensional pseudomanifold property is enough to guarantee the whole object is a well-defined discrete pseudomanifold without extra global consistency rules.

What would settle it

Exhibit either a discrete d-pseudomanifold on fewer than 2(d+1) vertices or a flag normal d-pseudomanifold on 2d+6 vertices that is not a simplicial sphere.

Figures

Figures reproduced from arXiv: 2606.29753 by Biplab Basak, Debolina Ghosh, Raju Kumar Gupta.

Figure 1
Figure 1. Figure 1: Some discrete 2-pseudomanifolds on 8 and 9 vertices. [PITH_FULL_IMAGE:figures/full_fig_p008_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Some discrete 3-pseudomanifolds W3 11a , W3 11b and W3 12 with 11 and 12 vertices Theorem 3.17. Let M be a discrete 3-pseudomanifold with 11 vertices. Then, up to iso￾morphism, M = M2 9 ⋆ S0 2 , where M2 9 is a discrete 2-pseudomanifold, or M = C6 ⋆ C5, or M = W3 11a , or M = W3 11b , where W3 11a and W3 11b are depicted in [PITH_FULL_IMAGE:figures/full_fig_p010_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Some discrete 2-pseudomanifolds on 10 vertices. [PITH_FULL_IMAGE:figures/full_fig_p013_3.png] view at source ↗
read the original abstract

A simple undirected graph $M$ is called a discrete $d$-pseudomanifold if, for every vertex $v$, the induced subgraph $N_M(v)$ on the neighbors of $v$ is a discrete $(d-1)$-pseudomanifold, where a discrete $1$-pseudomanifold is defined to be an $n$-cycle with $n\geq 4$. These objects arise naturally as graph-theoretic analogues of simplicial pseudomanifolds and provide a purely combinatorial framework for studying manifold-like structures through local neighborhood conditions. Understanding discrete pseudomanifolds with a small number of vertices is therefore a fundamental problem in combinatorial topology and extremal graph theory. In this article, we first prove that every discrete $d$-pseudomanifold has at least $2(d+1)$ vertices. We then provide a complete classification of discrete $d$-pseudomanifolds with at most $2d+6$ vertices by determining all possible combinatorial types of such pseudomanifolds. Further, we establish an equivalence between discrete $d$-pseudomanifolds and edge graphs of flag normal $d$-pseudomanifolds. As a consequence, we derive a purely combinatorial characterization of flag normal $d$-pseudomanifolds with at most $2d+6$ vertices and prove that each such complex is a simplicial $d$-sphere. Finally, we show that this sphere characterization is optimal within the class of flag normal $d$-pseudomanifolds by constructing examples on $2d+7$ vertices that are not spheres. Specifically, we prove that, for $d\geq 3$, every flag normal $d$-pseudomanifold with at most $2d+7$ vertices is either a simplicial $d$-sphere or a flag triangulation of the $(d-2)$-fold suspension of $\mathbb{RP}^{2}$.

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

1 major / 1 minor

Summary. The paper defines a discrete d-pseudomanifold recursively: a graph M is a discrete d-pseudomanifold if for every vertex v the induced subgraph N_M(v) is a discrete (d-1)-pseudomanifold (base case: n-cycle, n≥4). It proves every such object has at least 2(d+1) vertices, classifies all with ≤2d+6 vertices, shows equivalence to the edge graphs of flag normal d-pseudomanifolds, proves that flag normal d-pseudomanifolds with ≤2d+6 vertices are simplicial d-spheres, and shows optimality by proving that for d≥3 every flag normal d-pseudomanifold with ≤2d+7 vertices is either a simplicial d-sphere or a flag triangulation of the (d-2)-fold suspension of RP².

Significance. If the local recursive definition is shown to imply the standard global pseudomanifold axioms, the classification supplies explicit combinatorial types for small objects and a purely combinatorial characterization of small flag normal pseudomanifolds, which may be useful in extremal graph theory and combinatorial topology.

major comments (1)
  1. [Definition of discrete d-pseudomanifold] Definition (opening paragraph and recursive construction): the definition requires only that each induced neighbor subgraph N_M(v) is itself a discrete (d-1)-pseudomanifold. Standard pseudomanifold axioms also demand that the complex be pure of dimension d, that every (d-1)-face lies in exactly two d-faces, and that links are connected. No global incidence or connectivity axioms are stated, and the text does not contain an explicit lemma proving that the local recursive condition forces these global properties. Because the lower-bound proof, the enumeration up to 2d+6 vertices, the equivalence with flag normal pseudomanifolds, and the sphere claim all rely on the objects satisfying the full pseudomanifold structure, this gap is load-bearing for the central results.
minor comments (1)
  1. [Title and abstract] The title states classification up to 2d+7 vertices while the abstract states classification up to 2d+6 vertices and a separate optimality result at 2d+7; align the wording or add a clarifying sentence.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading and for highlighting the need to explicitly connect our recursive definition to the standard global pseudomanifold axioms. We agree this connection should be made fully explicit and will add the required material in revision.

read point-by-point responses

  1. Referee: [Definition of discrete d-pseudomanifold] Definition (opening paragraph and recursive construction): the definition requires only that each induced neighbor subgraph N_M(v) is itself a discrete (d-1)-pseudomanifold. Standard pseudomanifold axioms also demand that the complex be pure of dimension d, that every (d-1)-face lies in exactly two d-faces, and that links are connected. No global incidence or connectivity axioms are stated, and the text does not contain an explicit lemma proving that the local recursive condition forces these global properties. Because the lower-bound proof, the enumeration up to 2d+6 vertices, the equivalence with flag normal pseudomanifolds, and the sphere claim all rely on the objects satisfying the full pseudomanifold structure, this gap is load-bearing for the central results.

    Authors: We acknowledge that the manuscript presents the recursive definition without a dedicated lemma explicitly deriving the global pseudomanifold properties (purity, double incidence of codimension-1 faces, and link connectivity) from the local condition. These properties do follow by induction on dimension: the base case d=1 is an n-cycle (n≥4), which is a 1-pseudomanifold; assuming the claim for dimension d-1, each neighborhood N_M(v) is a discrete (d-1)-pseudomanifold and therefore satisfies the global axioms by the inductive hypothesis. The global structure on M then follows directly from the neighborhood conditions (every (d-1)-face lies in exactly two d-faces because its link in N_M(v) is a (d-2)-pseudomanifold, purity holds because neighborhoods are (d-1)-dimensional, and connectivity of links is inherited). The equivalence to flag normal d-pseudomanifolds is likewise proved in the paper and supplies the missing global axioms. Nevertheless, we agree an explicit lemma is desirable for clarity and will insert a new Lemma (after the definition) that records this inductive argument in full detail, together with a short corollary confirming that every discrete d-pseudomanifold is the 1-skeleton of a flag normal d-pseudomanifold. This revision will be marked as major. revision: yes

Circularity Check

0 steps flagged

No circularity; classification follows directly from recursive local definition

full rationale

The paper introduces a recursive definition of discrete d-pseudomanifolds via induced neighbor subgraphs and derives the vertex lower bound, complete classification up to 2d+6 vertices, equivalence to edge graphs of flag normal d-pseudomanifolds, and sphere characterization through direct combinatorial arguments. No equations, fitted parameters, self-referential predictions, or load-bearing self-citations appear. The recursive definition is inductive (base case n-cycle) rather than self-definitional, and the equivalence and optimality results at 2d+7 are presented as consequences rather than assumptions. The derivation chain is self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claims rest on the recursive neighborhood definition and the base-case cycle condition; no free parameters or invented entities are visible in the abstract.

axioms (1)
  • domain assumption A discrete 1-pseudomanifold is defined to be an n-cycle with n >= 4
    This base case is stated in the abstract and used to anchor the recursive definition of higher-dimensional discrete pseudomanifolds.

pith-pipeline@v0.9.1-grok · 5886 in / 1437 out tokens · 37259 ms · 2026-06-30T05:50:08.246781+00:00 · methodology

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Reference graph

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