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arxiv: 2507.02711 · v2 · submitted 2025-07-03 · 🧮 math.CO

A note on maximal plane subgraphs of the complete twisted graph containing perfect matchings

Elsa Oma\~na-Pulido , Eduardo Rivera-Campo This is my paper

Pith reviewed 2026-05-19 06:16 UTC · model grok-4.3

classification 🧮 math.CO
keywords twisted graphmaximal plane subgraphperfect matchingedge exchangegraph reconfigurationplane graphcomplete graph
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The pith

Any two maximal plane subgraphs of the twisted graph T_n with perfect matchings can be connected by a sequence of single edge exchanges.

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

The paper establishes a connectivity result for maximal plane subgraphs of the twisted graph T_n that each contain at least one perfect matching. It shows that for any two such subgraphs S and R, there exists a finite sequence of similar subgraphs starting at S and ending at R where each step replaces exactly one edge with another edge. All graphs in the sequence remain maximal plane and continue to contain a perfect matching. A reader would care because the result describes how these objects can be reconfigured locally without losing their key properties, under the specific crossing rules of T_n.

Core claim

For any maximal plane subgraphs S and R of T_n, each containing at least one perfect matching, there is a sequence S = F_0, F_1, …, F_m = R of maximal plane subgraphs of T_n, also containing perfect matchings, such that for i = 0 to m-1, F_{i+1} is obtained from F_i by a single edge exchange. The crossing condition of T_n, where two edges cross precisely when their four endpoints interleave in the stated index order, supports the existence of these exchanges at every step.

What carries the argument

Single edge exchange between maximal plane subgraphs of the twisted graph T_n that preserves both maximality and the existence of a perfect matching.

If this is right

  • Any such subgraph can reach any other through a finite chain of local replacements.
  • The set of all qualifying subgraphs forms a connected space under this move.
  • Transformations never require passing through a graph that loses planarity or its perfect matching.
  • The result applies uniformly for every n where T_n is defined.

Where Pith is reading between the lines

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

  • The same exchange method might apply to other drawings of complete graphs with controlled crossings.
  • One could check whether the connectivity still holds when requiring multiple distinct perfect matchings in each subgraph.
  • The exchanges provide a potential way to generate new examples starting from a known maximal plane subgraph.

Load-bearing premise

The particular crossing rule of the twisted graph T_n allows every needed single edge exchange to keep the current subgraph maximal and still containing a perfect matching.

What would settle it

Two maximal plane subgraphs of T_n for some n, each with a perfect matching, such that no sequence of single edge exchanges connects them while keeping every intermediate graph maximal plane and matched.

Figures

Figures reproduced from arXiv: 2507.02711 by Eduardo Rivera-Campo, Elsa Oma\~na-Pulido.

Figure 1
Figure 1. Figure 1: The twisted graph T6. which is weakly isomorphic to either the complete convex geometric graph Cm or to the twisted graph Tm. Let Fn be a complete topological graph with n vertices. A plane subgraph G of Fn is a maximal plane subgraph of Fn if for each edge uv of Fn, not in G, there is an edge xy of G that intersects uv. The max graph MP (Fn) of Fn is the abstract graph whose vertices are the maximal plane… view at source ↗
Figure 2
Figure 2. Figure 2: Graphs R (above) and S1 (bellow). in the argument in Case k(R, Q) = 0, we obtain a path R, S′ 1 , S′ 2 , . . . , S′ l ′ in MP(Tn, F) that joins R and S ′ l ′, where l ′ = maxR(vk+1) − maxQ(vk+1) and dS ′ l ′ (vk+1) = dQ(vk+1). Then k(S ′ l ′, Q) ≥ k(R, Q)+1 and m(S ′ l ′, Q) ≤ t. By the Induction Hypothesis MP(Tn, F) contains a path joining S ′ l ′ and Q which, together with the path R, S′ 1 , S′ 2 , . . .… view at source ↗
Figure 3
Figure 3. Figure 3: A maximal plane subgraph ot T6 with no perfect matchings. Theorem 3. For each positive integer m, the graph MPM(T2m) is connected. 4 [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
read the original abstract

The twisted graph $T_{n}$ is a drawing of the complete graph with $n$ vertices $v_{1},v_{2},\ldots ,v_{n}$ in which two edges $v_{i}v_{j}$ ($i<j$) and $v_{s}v_{t}$ ($s<t$) cross if and only if $i<s<t<j$ or $s<i<j<t$. We show that for any maximal plane subgraphs $S$ and $R$ of $T_{n}$, each containing at least one perfect matching, there is a sequence $S=F_0, F_1, \ldots, F_m=R$ of maximal plane subgrahs of $T_n$, also containing perfect matchings, such that for $i=0,1, \ldots, m-1$, $F_{i+1}$ can be obtained from $F_{i}$ by a single edge exchange.

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 / 2 minor

Summary. The manuscript defines the twisted graph T_n as a drawing of the complete graph K_n on vertices v1 to vn in convex position, where two edges vi vj (i<j) and vs vt (s<t) cross precisely when i < s < t < j or s < i < j < t. It proves that any two maximal plane subgraphs S and R of T_n, each containing at least one perfect matching, are connected by a finite sequence S = F0, F1, …, Fm = R of maximal plane subgraphs that also contain perfect matchings, with each consecutive pair differing by exactly one edge exchange.

Significance. If the result holds, it establishes connectivity of the reconfiguration graph on maximal plane subgraphs with perfect matchings under single edge exchanges. This supplies an explicit, constructive transformation between any two such subgraphs and may inform broader questions on the structure of plane matchings in geometric graphs with controlled crossings. The proof proceeds by case analysis on relative vertex positions under the given crossing rule and explicitly constructs the required alternating-path rerouting to preserve a perfect matching at each step; this case-by-case verification is a clear strength.

major comments (1)
  1. [Abstract and opening paragraph of the proof] The central claim is stated for general n, yet perfect matchings exist only for even n. The case analysis is described as covering all even-n configurations; the manuscript should explicitly restrict the statement to even n (or add the odd-n case as vacuous) so that the hypothesis is well-formed.
minor comments (2)
  1. [Abstract] Typo in the abstract: 'subgrahs' should read 'subgraphs'.
  2. [Introduction] The crossing condition is given twice (once in the abstract, once in the introduction); a single, numbered definition would improve readability.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading of the manuscript and for the positive assessment of the result. We agree with the observation regarding the domain of n and will revise the manuscript to make the statement precise.

read point-by-point responses

  1. Referee: [Abstract and opening paragraph of the proof] The central claim is stated for general n, yet perfect matchings exist only for even n. The case analysis is described as covering all even-n configurations; the manuscript should explicitly restrict the statement to even n (or add the odd-n case as vacuous) so that the hypothesis is well-formed.

    Authors: We thank the referee for this observation. Perfect matchings exist only for even n, and the case analysis in the proof already proceeds under that assumption. We will revise the abstract and the opening paragraph of the proof to state explicitly that n is even (or, equivalently, to note that the claim is vacuous for odd n). This is a minor clarification that does not affect the substance of the argument or the constructions. revision: yes

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The paper establishes an existence result for transforming one maximal plane subgraph with a perfect matching into another via single edge exchanges in the twisted graph T_n. The argument proceeds by explicit case analysis on vertex indices under the paper's stated crossing condition (i < s < t < j or s < i < j < t), constructing the required swaps while preserving planarity, maximality, and the existence of a perfect matching through alternating paths. No parameters are fitted, no quantities are defined in terms of the result being proved, and the derivation does not invoke self-citations or prior results by the same authors as load-bearing steps. The proof is self-contained within the combinatorial definitions and case distinctions given in the manuscript.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper rests on the standard definition of plane subgraphs and perfect matchings together with the explicit crossing rule for T_n. No free parameters, new entities, or ad-hoc axioms beyond domain assumptions of geometric graph theory are visible in the abstract.

axioms (2)
  • domain assumption Two edges vi vj (i<j) and vs vt (s<t) cross if and only if i<s<t<j or s<i<j<t.
    This crossing rule is the defining property of the twisted graph T_n invoked in the statement.
  • domain assumption A subgraph is maximal plane if no additional edge from T_n can be added without creating a crossing.
    Standard definition used to characterize the objects whose connectivity is claimed.

pith-pipeline@v0.9.0 · 5699 in / 1496 out tokens · 73109 ms · 2026-05-19T06:16:25.965484+00:00 · methodology

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Reference graph

Works this paper leans on

5 extracted references · 5 canonical work pages

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    work page 1992

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    Oma˜ na-Pulido, E., Rivera-Campo, E.: Notes on the twisted graph, in M´ arquez, Alberto (ed.) et al., Computational Geometry, Lecture Notes in Computer Science 7579, (2012), 119 - 125

    work page 2012

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    Pach, J., Solymosi, J., T´ oth, G.: Unavoidable configurations in complete topological graphs, Discrete Comput. Geom. 30, 311–320 (2003). 5

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