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arxiv: 1907.06951 · v1 · pith:U5MJ4H45new · submitted 2019-07-16 · 🧮 math.CO

on removal of perfect matching from folded hypercubes

S.A.Mane This is my paper

Pith reviewed 2026-05-24 20:55 UTC · model grok-4.3

classification 🧮 math.CO
keywords folded hypercubeperfect matchinghypercubegraph isomorphismconjecturecounterexampleinterconnection network
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The pith

Removing some perfect matchings from folded hypercubes does not produce a graph isomorphic to the hypercube.

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

The paper addresses a conjecture claiming that a subset of edges Em forms a perfect matching in the folded hypercube FQn precisely when the graph obtained by removing Em is isomorphic to the n-dimensional hypercube Qn. It constructs explicit counterexamples consisting of perfect matchings in FQn for which the removal leaves a graph that is not isomorphic to Qn. This corrects an incorrect characterization of the relationship between these two families of graphs. The result matters for the study of interconnection networks because it shows that the structural properties allowing perfect matchings to recover hypercubes do not hold universally in the folded case.

Core claim

Dong and Wang conjectured that a subset Em of edges of FQn is a perfect matching if and only if FQn minus Em is isomorphic to Qn. This paper disproves the conjecture by providing some perfect matchings whose removal from FQn does not give a graph isomorphic to Qn.

What carries the argument

Explicit constructions of perfect matchings Em in the folded hypercube FQn such that FQn - Em is not isomorphic to the hypercube Qn.

Load-bearing premise

The specific perfect matchings presented in the paper are valid perfect matchings of FQn and the graphs obtained after their removal are verifiably not isomorphic to Qn.

What would settle it

A direct verification showing that any one of the paper's example perfect matchings in FQn, when removed, actually produces a graph isomorphic to Qn would falsify the disproof.

read the original abstract

The hypercube Qn of dimension n is one of the most versatile and powerful interconnection networks. The n-dimensional folded cube denoted as FQn, a variation of the hypercube possesses some embeddable properties that the hypercube does not possess. Dong and Wang(In Theor. Comput. Sci.771(2019)93-98) conjectured that "A subset Em of edges of FQn is a perfect matching if and only if FQn - Em is isomorphic to Qn". In this paper, we disprove this conjecture by providing some perfect matchings removal of which from FQn do not give a graph isomorphic to Qn.

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 / 0 minor

Summary. The paper disproves the conjecture of Dong and Wang that a subset Em of edges of the folded hypercube FQn is a perfect matching if and only if FQn - Em is isomorphic to Qn, by exhibiting explicit perfect matchings whose removal yields graphs that are not isomorphic to the n-dimensional hypercube Qn.

Significance. If the counterexamples are valid, the result falsifies the stated if-and-only-if claim and clarifies the relationship between perfect matchings in folded hypercubes and the resulting graphs' isomorphism type. Explicit counterexamples constitute a direct and falsifiable method of disproof in this area of graph theory and interconnection networks.

major comments (1)
  1. [constructions of the counterexamples] The manuscript asserts the existence of counterexamples but does not supply explicit edge listings for the perfect matchings Em, nor does it detail the verification that each Em is a perfect matching (every vertex incident to exactly one edge) or that FQn - Em fails an isomorphism invariant of Qn. This verification is load-bearing for the central claim of disproving the conjecture.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the detailed review and for identifying the need for greater explicitness in our counterexamples. We address the major comment below and will revise the manuscript to strengthen the presentation of the disproof.

read point-by-point responses

  1. Referee: [constructions of the counterexamples] The manuscript asserts the existence of counterexamples but does not supply explicit edge listings for the perfect matchings Em, nor does it detail the verification that each Em is a perfect matching (every vertex incident to exactly one edge) or that FQn - Em fails an isomorphism invariant of Qn. This verification is load-bearing for the central claim of disproving the conjecture.

    Authors: We agree that the current manuscript would benefit from more explicit constructions. While the paper describes families of perfect matchings whose removal yields non-isomorphic graphs, the edge sets are presented at a level of generality that does not include concrete listings or step-by-step verification for specific n. In the revised version we will add, for at least two small values of n (e.g., n=4 and n=5), complete lists of the edges in Em, a direct check that each vertex is incident to exactly one edge of Em, and an explicit computation of an isomorphism invariant (such as the number of 4-cycles or the diameter) that distinguishes FQn − Em from Qn. revision: yes

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The paper disproves the Dong-Wang conjecture via explicit counterexamples consisting of perfect matchings Em in FQn such that FQn − Em fails to be isomorphic to Qn. No derivation, ansatz, fitted parameter, or self-citation chain is present; the central claim is established by direct construction and verification of the listed matchings and the resulting graphs' failure to satisfy isomorphism invariants of Qn. The argument is therefore self-contained against external graph-theoretic benchmarks with no reduction of outputs to inputs by the paper's own definitions or equations.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The work rests on standard graph-theoretic definitions of hypercubes and folded hypercubes plus the correctness of the claimed counterexamples; no free parameters or invented entities are introduced.

axioms (2)
  • standard math Standard definitions of the n-dimensional hypercube Qn and folded hypercube FQn as graphs with their edge sets.
    The conjecture and its disproof are stated in terms of these established objects.
  • domain assumption The provided examples constitute perfect matchings whose removal yields non-isomorphic graphs.
    The disproof depends on the validity of these specific constructions and isomorphism checks.

pith-pipeline@v0.9.0 · 5621 in / 1238 out tokens · 23347 ms · 2026-05-24T20:55:34.386568+00:00 · methodology

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