pith. sign in

arxiv: 2606.17605 · v2 · pith:DSK3KRZBnew · submitted 2026-06-16 · 🧮 math.CO · math.GR· math.MG

A coarse Menger theorem for hyperbolic graphs, finitely presented groups, and more

Sandra Albrechtsen This is my paper

Pith reviewed 2026-06-27 00:29 UTC · model grok-4.3

classification 🧮 math.CO math.GRmath.MG
keywords coarse Menger propertycycle spacehyperbolic graphsfinitely presented groupsMenger's theoremplanar graphsgeodesic metric spacesbounded length cycles
0
0 comments X

The pith

Graphs whose cycle space is generated by cycles of bounded length have the coarse Menger property.

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

The paper shows that graphs in which every cycle is a sum of cycles of some fixed bounded length possess the coarse Menger property. This is a relaxed form of Menger's theorem in which the paths between two sets must be pairwise far apart and the separator consists of a small number of bounded-radius balls rather than single vertices. The result immediately yields the property for hyperbolic graphs, Cayley graphs of finitely presented groups, planar graphs with bounded faces, and complete Riemannian planes. A sympathetic reader cares because the bounded-cycle-generation condition supplies a single structural hypothesis that recovers a basic connectivity statement across many geometrically natural classes.

Core claim

If the cycle space of a graph is generated by cycles of bounded length, then for any vertex sets X and Y, any positive integer k, and any distance d, either there exist k paths from X to Y that are pairwise at distance at least d, or there exists a set of at most k-1 balls of bounded radius that intersects every X-Y path.

What carries the argument

The cycle space generated by cycles of bounded length, which supplies short cycles that bound the radius of the balls needed for separators.

Load-bearing premise

The cycle space of the graph is generated by cycles of bounded length.

What would settle it

An explicit graph whose cycle space is generated by cycles of length at most some fixed L, together with sets X, Y, integers k and d, such that neither k far-apart paths nor a (k-1)-ball separator of bounded radius exists.

Figures

Figures reproduced from arXiv: 2606.17605 by Sandra Albrechtsen.

Figure 1
Figure 1. Figure 1: Depicted are paths Pi , pairwise at distance at least D. The components C of G − S BG(Pi , r)  , for r << D, connect the sets BG(Pi , r) in a ‘tree-like’ way: The hypergraph with vertex set {P1, . . . , Pk−1} in which the components C are turned into hyper-edges is acyclic (see right corner). Moreover, for every Pi and every component C attaching to BG(Pi , r), the  κ–2 2  -neighbourhood of BG(Pi , r) i… view at source ↗
Figure 2
Figure 2. Figure 2: By the induction hypothesis, we find k − 1 X–Y paths Pi that are pairwise at distance at least D. Lemma 4.2 yields for every Pi some small-radius ball Bi (depicted in orange) and two X–Bi paths (in blue) that are pairwise at distance at least d, one of which avoids all other BG(Pj , r). If the Bi do not hit all X–Y paths, then there is a component C that attaches ‘before’ some Bi and ‘behind’ some Bj (in g… view at source ↗
Figure 3
Figure 3. Figure 3: An illustration of Lemma 4.2: There are either two X′–Y ′ paths as in (i), which are at distance at least d and which are (almost) contained in BG [PITH_FULL_IMAGE:figures/full_fig_p010_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: An illustration of the situation in the proof of [PITH_FULL_IMAGE:figures/full_fig_p013_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Illustration of the paths P ′ 2 , Q′ 1 (left) and P ′ ℓm+1 , Q′ m+1 (right). On the right side, i = 2. Construction of P′ 2 = P′ ℓ1 and Q′ 1 : (See [PITH_FULL_IMAGE:figures/full_fig_p015_5.png] view at source ↗
read the original abstract

Menger's theorem is one of the most fundamental results in graph theory. It states that if a graph $G$ does not contain $k$ disjoint paths between two given sets $X$ and $Y$ of vertices in $G$, then there is a set of at most $k-1$ vertices that intersects every path between $X$ and $Y$. Nguyen, Scott, and Seymour gave a counterexample to the conjectured natural coarse variant in which the paths are required to be pairwise at distance at least $d$, and, conversely, there is a set of at most $k-1$ bounded-radius balls intersecting every path between $X$ and $Y$. In other words, the coarse Menger property does not hold in general. We prove that graphs whose cycles space is generated by cycles of bounded length do have the coarse Menger property. As a corollary, we show that many natural graphs and geodesic metric spaces have the coarse Menger property. These include hyperbolic graphs, Cayley graphs of finitely presented groups, planar graphs with bounded face size, and complete Riemannian planes.

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 / 3 minor

Summary. The manuscript proves that graphs whose cycle space is generated by cycles of bounded length satisfy the coarse Menger property: if there do not exist k pairwise distance-at-least-d paths between vertex sets X and Y, then there exists a separator consisting of at most k-1 balls of bounded radius that intersects every X-Y path. Corollaries are derived for hyperbolic graphs, Cayley graphs of finitely presented groups, planar graphs with bounded face size, and complete Riemannian planes, by verifying that these classes satisfy the cycle-space hypothesis.

Significance. If the central theorem holds, the result supplies a natural, checkable combinatorial condition (bounded-length cycle-space generators) that is sufficient for the coarse Menger property and that is known to hold for several geometrically significant classes. This directly addresses the counterexample of Nguyen-Scott-Seymour by delineating a broad family of graphs and spaces in which the coarse variant is valid, with potential utility in coarse geometry and geometric group theory.

minor comments (3)
  1. Abstract, line 3: 'cycles space' should read 'cycle space'.
  2. The main theorem statement (presumably §3 or §4) would benefit from an explicit remark on whether the radius bound in the separator is a function of the cycle-length bound alone or also depends on other graph parameters.
  3. In the corollaries section, the verification that each listed class has its cycle space generated by bounded-length cycles should include a brief pointer to the relevant reference or short argument establishing the bound, to make the applications self-contained.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the careful reading, positive assessment of the significance, and recommendation of minor revision. No major comments were listed in the report.

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained

full rationale

The paper states a conditional theorem: graphs whose cycle space is generated by cycles of bounded length satisfy the coarse Menger property. This is presented as a direct proof under an explicitly declared hypothesis, with corollaries obtained by verifying that standard classes (hyperbolic graphs, finitely presented groups, etc.) meet the hypothesis. No equations, fitted parameters, self-citations, or ansatzes are invoked in the abstract or described argument that reduce the conclusion to the input by construction. The result is therefore independent of its own statement.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Only the abstract is available, so the ledger records the minimal structural assumptions visible; no free parameters, invented entities, or non-standard axioms are stated.

axioms (1)
  • standard math Standard facts from graph theory and algebraic topology on cycle spaces (e.g., that the cycle space is a vector space over GF(2)).
    Invoked implicitly when the paper refers to 'cycle space is generated by cycles of bounded length'.

pith-pipeline@v0.9.1-grok · 5726 in / 1191 out tokens · 32630 ms · 2026-06-27T00:29:57.016815+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

12 extracted references · 4 linked inside Pith

  1. [1]

    Small counterexamples to the fat minor conjecture

    arXiv: 2508.15342. [ADG26] S. Albrechtsen, M. Distel, and A. Georgakopoulos. “Small counterexamples to the fat minor conjecture”

    arXiv

  2. [2]

    Asymptotic half-grid and full-grid minors

    arXiv:2601.05761. [AH26] S. Albrechtsen and M. Hamann. “Asymptotic half-grid and full-grid minors”. In:Journal of Combinatorical Theory, Series B176 (2026), pp. 440–485. [AHJ+24] S. Albrechtsen, T. Huynh, R. W. Jacobs, P. Knappe, and P. Wollan. “A Menger-type theorem for two induced paths”. In:SIAM Journal on Discrete Mathematics38.2 (2024), pp. 1438–1450...

    arXiv 2026

  3. [3]

    Fat minors cannot be thinned (by quasi-isometries)

    arXiv:2605.11112. [DHIM24] J. Davies, R. Hickingbotham, F. Illingworth, and R. McCarty. “Fat minors cannot be thinned (by quasi-isometries)”

    Pith/arXiv arXiv

  4. [4]

    String graphs are quasi-isometric to planar graphs

    arXiv:2405.09383. [Dav25] J. Davies. “String graphs are quasi-isometric to planar graphs”

    arXiv

  5. [5]

    [Die24] R

    arXiv:2510.19602. [Die24] R. Diestel. Graph Theory(6th edition). Springer-Verlag,

    arXiv

  6. [6]

    Infinite-ended groups with planar Cayley graphs

    arXiv:2509.08762v2. 18 Sandra Albrechtsen [Dro06] C. Droms. “Infinite-ended groups with planar Cayley graphs”. In:Journal of Group Theory9 (2006), pp. 487–496. [EGG26] L. Esperet, H. Gahlawat, and U. Giocanti. “Coarse cops and robber in graphs and groups”. In: European Journal of Combinatorics135 (2026), p. 104356. [GKL23] P. Gartland, T. Korhonen, and D....

    Pith/arXiv arXiv 2006

  7. [7]

    Graph minors and metric spaces

    arXiv:2309.08169. [GP25] A. Georgakopoulos and P. Papasoglu. “Graph minors and metric spaces”. In:Combinatorica 45.33 (2025). [Ham18a] M. Hamann. “Accessibility in Transitive Graphs”. In:Combinatorica 38 (2018), pp. 847–859. [Ham18b] M. Hamann. “Planar transitive graphs”. In:Electronic Journal of Combinatorics25(4) (2018), Article No. P4.8. [HNST24] K. He...

    arXiv 2025

  8. [8]

    Almost planar finitely presented groups

    arXiv: 2605.10068. [MMS26] J. M. Mackay, J. P. MacManus, and D. Spriano. “Almost planar finitely presented groups”

    Pith/arXiv arXiv

  9. [9]

    Fat minors in finitely presented groups

    arXiv: 2605.03040. [Mac25] J. P. MacManus. “Fat minors in finitely presented groups”. In:Combinatorica 45.40 (2025). [Men27] K. Menger. “Zur allgemeinen Kurventheorie”. In:Fundamenta Mathematicae10 (1927), pp. 96–

    Pith/arXiv arXiv 2025

  10. [10]

    A counterexample to the coarse Menger conjecture

    [NSS25a] T. Nguyen, A. Scott, and P. D. Seymour. “A counterexample to the coarse Menger conjecture”. In: Journal of Combinatorial Theory, Series B173 (2025), pp. 68–82. arXiv:2401.06685. [NSS25b] T. Nguyen, A. Scott, and P. D. Seymour. “Asymptotic structure. IV. A counterexample to the weak coarse Menger conjecture”

    arXiv 2025

  11. [11]

    Asymptotic structure. VI. Distant paths across a disc

    arXiv:2508.14332. [NSS25c] T. Nguyen, A. Scott, and P. D. Seymour. “Asymptotic structure. VI. Distant paths across a disc”

    arXiv

  12. [12]

    arXiv:2509.07174. 19

    arXiv