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Tropically planar graphs

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Pith's one-line read The paper proves that tropically planar graphs are asymptotically 0% of connected trivalent planar graphs, with explicit upper and lower bounds on their number.

desk verdict Solid new upper bound and a plausible lower bound that needs computational verification; the zero-density result is correct despite a gap in one of its proofs. read the letter →

arxiv 1908.04320 v3 pith:KS6GT7IF submitted 2019-08-12 math.AG math.CO

Desmond Coles , Neelav Dutta , Sifan Jiang , Ralph Morrison , Andrew Scharf This is my paper · ORCID
classification math.AGmath.CO MSC 14T0505C3005C1052B20
keywords tropicallyplanargraphstropicalplanecurvesskeletonofacurveregularunimodulartriangulationslatticepolygonstrivalentasymptoticenumerationwidth
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

Tropically planar graphs are the graphs that appear as skeletons of smooth tropical plane curves. This paper proves that, as the genus grows, these graphs become vanishingly rare among connected trivalent planar graphs: the ratio $T(g)/P(g)$ tends to 0. It establishes quantitative bounds, $T(g)=O(2^{11g/3+O(\sqrt g)})$ and $T(g)=\Omega(\gamma^g)$ with $\gamma\approx 2.47$, and it extends the exact enumeration from genus 5 to genus 6 and 7, finding 152 and 672 tropically planar graphs respectively. Along the way it develops new necessary conditions, including a forbidden TIE-fighter configuration and a bridge-reduction surgery, that rule out many non-tropically planar graphs.

What carries the argument

The argument runs on three engines. First, the duality between a smooth tropical plane curve and a regular unimodular triangulation of its Newton polygon: each troplanar graph is the skeleton dual to such a triangulation, so counting troplanar graphs becomes counting regular unimodular triangulations of lattice polygons of genus $g$ up to the graphs they yield. Second, an upper-bound engine that stratifies polygons by lattice width $\ell$: width-2 (hyperelliptic) polygons contribute $O(2^g)$ graphs, width-3 polygons contribute $O(8^g\sqrt g)$ each via the binomial coefficient $\binom{g-2}{a}$ of unimodular triangulations of the interior trapezoid, and width-at-least-4 polygons are bounded using $r\le 2g/\ell+4\sqrt{g+8/3}+2$ boundary points together with the general triangulation bound $2^{3g+r-3}$. Third, a lower-bound engine that tiles the parallelogram $P^{\parallel}_{2n}$ with 2 tiles of genus 2, 13 of genus 4, and 75 of genus 6; regularity is preserved when gluing along lattice-length-1 edges, and Proposition 5.1 ensures different tile sequences yield non-isomorphic graphs, producing the recurrence $a_n=2a_{n-1}+13a_{n-2}+75a_{n-3}$ whose dominant root $\alpha\approx 6.1233$ gives $\gamma=\sqrt{\alpha}\approx 2.47$. A supporting surgery, bridge reduction, uses a bistellar flip to show that reducing all bridges of a troplanar graph yields a 2-edge-connected troplanar graph, giving $T(g)\le 2^{g-1}T^{(2)}(g)$.

What would settle it

Independently enumerate all regular unimodular triangulations of the parallelograms $P^{\parallel}_2$, $P^{\parallel}_4$, and $P^{\parallel}_6$ and compare the resulting marked skeletons against the paper's Appendix A tiles. A duplicate marked graph, a tile that is not a regular triangulation, or a genus-6 or genus-7 total different from 152 or 672 would force the corresponding bound or count to be revised downward.

Watch

Extended reading notes

Core claim

On the paper's own terms, the central discovery is that troplanar graphs are exponentially rare among all connected trivalent planar graphs: $\lim_{g\to\infty} T(g)/P(g)=0$, with the quantitative bounds $T(g)=O(2^{11g/3+O(\sqrt g)})$ and $T(g)=\Omega(\gamma^g)$, where $\gamma=\sqrt{\alpha}\approx 2.47$ and $\alpha$ is the unique real root of $x^3-2x^2-13x-75$. The upper bound comes from stratifying lattice polygons by lattice width and bounding the number of their unimodular triangulations; the lower bound comes from tiling a genus-$2n$ parallelogram with fixed tiles of genus 2, 4, and 6, giving a recurrence whose growth rate is governed by $\alpha$. The paper also settles the exact enumeration through genus 7: $T(6)=152$ and $T(7)=672$, continuing the sequence 2, 4, 13, 38, 152, which does not appear in the OEIS. These results supersede the previously known lower bound $T(g)=\Omega(2^g)$ from hyperelliptic chains.

Load-bearing premise

The lower bound assumes that the 75 genus-6 tiles and 13 genus-4 tiles listed in the appendix are all distinct as marked graphs and all realizable as regular triangulations of the parallelogram; if any two are actually the same, or one fails regularity, the tiling count and the base $\gamma\approx 2.47$ would shrink.

Editorial extensions

If this is right

  • If the bounds are correct, troplanar graphs are not just a minority but an exponentially negligible proportion of connected trivalent planar graphs: the ratio $T(g)/P(g)$ tends to 0 at an exponential rate.
  • The exact counts $T(6)=152$ and $T(7)=672$ give the first data beyond genus 5, and the sequence 2, 4, 13, 38, 152 matches no OEIS sequence, so any proposed formula for $T(g)$ must reproduce these values.
  • The upper bound $O(2^{11g/3+O(\sqrt g)})$ improves on the generic planar-graph bound (base roughly 15.88) and shows the exponential base of $T(g)$ is at most $2^{11/3}\approx 12.7$.
  • The lower bound $\Omega(\gamma^g)$ with $\gamma\approx 2.47$ supersedes the hyperelliptic-chain lower bound $\Omega(2^g)$, so the true exponential base lies somewhere in $[2.47,12.7]$.
  • New necessary conditions, namely that TIE-fighter graphs are never troplanar and that bridge reduction preserves troplanarity, give practical tests for ruling graphs in or out at any fixed genus.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • If regular triangulations are as rare among all unimodular triangulations as numerical experiments suggest, the true exponential base of $T(g)$ could be much closer to the lower bound 2.47 than to the upper bound 12.7; this is an extrapolation, not a paper claim.
  • Extending the tiling construction to tiles of genus 8 or higher would change the recurrence's characteristic polynomial, and the growth base of the lower bound is the dominant root of that polynomial, so each new tile family is a lever for raising $\gamma$; the authors mention this direction only in passing.
  • The paper notes that proving a 'two loops in a row' obstruction analogous to Corollary 3.5 would remove 18 of the 28 genus-6 graphs not ruled out by any known criterion; supplying that proof is a direct way to test whether $T(6)=152$ can be improved.
  • Because the upper bound counts all unimodular triangulations rather than only regular ones, a sharper census of regular triangulations of lattice polygons could both validate the lower-bound tile list and narrow the gap between 2.47 and 12.7.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 7 minor

Summary. The paper studies trophically planar graphs, defined as skeletons of smooth tropical plane curves. It develops new necessary conditions for a graph to be trophically planar (notably the TIE-fighter obstruction), computes the exact numbers of trophically planar graphs of genus 6 and 7 (152 and 672), and proves asymptotic estimates: an upper bound T(g) = O(2^{11g/3+O(√g)}) and a lower bound T(g) = Ω(γ^g) with γ ≈ 2.47. From these results the authors conclude that, asymptotically, 0% of connected trivalent planar graphs are trophically planar.

Significance. If the claims hold, the paper makes a substantial contribution to the quantitative study of trophically planar graphs. The upper bound is a nontrivial exponential estimate, and the lower bound improves the previously known Ω(2^g) to a base γ ≈ 2.47, a genuine advance. The zero-density statement is a strong negative result. The TIE-fighter obstruction is an elegant new structural tool, and the exact counts for genus 6 and 7 provide valuable data. The paper also includes a large set of computational tiles, but the reproducibility of that finite verification is a major weakness, as detailed below.

major comments (2)
  1. [Section 5, Corollary 5.5] The lower bound rests on an unverified finite tile inventory. The paper asserts that a TOPCOM search shows that the parallelograms P^||_2, P^||_4, and P^||_6 admit no non-regular triangulations, and that the 2+13+75 pictured tiles are distinct marked graphs, but it supplies no code, input files, output logs, or the actual triangulations. Appendix A presents only the marked skeletons, not the triangulations of the parallelograms. A duplicate or non-regular tile would change the coefficients of the recurrence a_n = 2a_{n-1} + 13a_{n-2} + 75a_{n-3} in Proposition 5.4 and could lower γ below the claimed value of √α ≈ 2.47, potentially even below the previously known Ω(2^g) bound. The lower-bound theorem is therefore conditional on a finite verification that is not made reproducible.
  2. [Section 5, Proposition 5.3] The proof of Proposition 5.3 relies on two unproved assertions about the tile set: that no two tiles give the same ordered pair of marked graphs, and that no tile contributing two 2-edge-connected components contributes a component that is also available from a single-tile component. These are nontrivial combinatorial properties over a set of 90 tiles. The paper neither proves them nor provides a machine-checkable certificate. Since the injectivity of the construction (and hence the lower bound on T(g)) depends on these properties, they should be verified explicitly, for example by a table of the ordered pairs or a reproducible script.
minor comments (7)
  1. [Section 2.3] There is a typo in the sentence 'a tropical curves has one vertex for each subpolygon in the subdivision'; it should read 'a tropical curve has'.
  2. [Section 3] The sentence 'It is not always immediately obvious is a graph if crowded' is grammatically garbled; it should likely read 'It is not always immediately obvious whether a graph is crowded.'
  3. [Section 5, proof of Proposition 5.2] In the last paragraph, the text says 'H1,··· ,Hm are the graphs arising from the tiles T′_1,··· ,T′_k' but the index should be m, not k.
  4. [Section 5, proof of Corollary 5.5] There is a duplicated word: 'we have have T(g) ≥ ...' should read 'we have T(g) ≥ ...'.
  5. [Section 5, Proposition 5.4] The derivation of the asymptotic lower bound uses numerical approximations A ≈ 0.49999, B ≈ 0.25001 + 0.00543i, α ≈ 6.1233, and r ≈ 3.4998. Since the conclusion a_n = Ω(α^n) requires exact inequalities such as A > 0 and α > r, the authors should replace these floating-point approximations with rigorous bounds, e.g. via interval arithmetic or explicit algebraic estimates.
  6. [Section 4, Theorem 4.2] The claim that any trivalent graph containing a copy of H is sprawling is not immediate because H contains a degree-1 vertex and the copy need not be induced. The statement is true, since all other vertices of H have degree 3 in H and hence are saturated in any ambient trivalent graph, but a brief justification would make the proof self-contained.
  7. [Section 5] The markings L and R are used throughout the construction, but the paper does not formally define a 'marked graph' or what it means for two marked graphs to be isomorphic. A precise definition would improve rigor and readability.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the exact counts and upper/lower bounds are derived from independent computer enumeration, external structural results, and a combinatorial tiling argument, none of which reduce to the paper's target data.

full rationale

The paper's derivation chain does not reduce to its inputs by construction. The exact counts T(6)=152 and T(7)=672 are obtained by enumerating regular unimodular triangulations of maximal genus-6 and genus-7 polygons with TOPCOM and then computing dual skeletons; this is a computation, not a fit to known values. The upper bound in Theorem 4.14 rests on external results: Bodirsky-Kang-Löffler-McDiarmid and Noy-Requile-Rue for random cubic planar graphs, De Loera-Rambau-Santos for triangulation counts, and Castryck's polygon bounds; the lattice-width-2 contribution is an explicit formula from prior work and only contributes O(2^g), so even if that cited formula were wrong, the exponential base 2^{11g/3} would be unchanged. The lower bound in Section 5 is a combinatorial tiling construction: the 2 genus-2 tiles, 13 genus-4 tiles, and 75 genus-6 tiles produce the recurrence a_n = 2a_{n-1} + 13a_{n-2} + 75a_{n-3}, and Proposition 5.1 gives a genuine graph-theoretic argument that distinct tile sequences yield non-isomorphic graphs. The tile inventory is input data verified by TOPCOM and by inspection of the pictures, not data fitted to T(g). The self-citations to [6] and [21], which share author Morrison, are used as prior published results (maximal-polygon reduction, crowded criterion, hyperelliptic skeleton counts) with independent content, and they are not restatements of the present paper's new claims. The main legitimate concern is reproducibility, not circularity: the TOPCOM search is described without code or logs, and the distinctness of the pictured marked tiles is asserted rather than formally proved. Those gaps affect correctness verification, but no equation in the paper is equivalent to its own input by definition and no fitted parameter is renamed as a prediction.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

No new free parameters are fitted; constants such as alpha, gamma, C1, and C2 arise from recurrences or cited theorems. The main domain assumptions are the standard tropical duality, the maximal polygon reduction, the regularity-patching lemma, and correctness of the TOPCOM computations. The lower-bound construction introduces 'tiles' as graph-building devices, but these are not newly postulated physical or mathematical entities with independent falsifiable handles.

assumptions (6)
  • standard math Pick's theorem relates area, boundary lattice points, and interior lattice points.
    Used throughout to convert area bounds to genus bounds and to identify unimodular triangles; introduced in Theorem 2.1.
  • domain assumption Duality between tropical plane curves and regular subdivisions of Newton polygons, with smoothness equivalent to unimodular triangulations.
    This is the foundational dictionary from tropical geometry, cited to [20] and used in Definitions 1.1 and Section 2.3.
  • domain assumption Every maximal nonhyperelliptic polygon is obtained by moving out the edges of its interior polygon.
    From [19, Lemma 2.2.13], used in Proposition 3.9 to classify maximal polygons of genus 6 and 7.
  • domain assumption Patching regular triangulations along edges of lattice length 1 preserves regularity, and all unimodular triangulations of hyperelliptic polygons are regular.
    From [18, Proposition 3.4], used in the lower-bound construction to guarantee that the tilings give regular triangulations.
  • domain assumption Asymptotic growth and concentration results for random cubic planar graphs, including the fixed-subgraph concentration theorem and the exponential growth rate of P(g).
    Cited from [5] and [22]; used in Theorem 4.2 and Corollary 4.4 to compare T(g) with P(g).
  • domain assumption The TOPCOM software correctly computes all regular unimodular triangulations of the input polygons and tile configurations.
    The exact counts and tile realizability depend on this external computational tool, whose inputs are not shipped.

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Pith. "Pith review of Tropically planar graphs." pith.science (2026). https://pith.science/paper/KS6GT7IF

@misc{pith2026190804320,
  author       = {Pith},
  title        = {Pith review of: Tropically planar graphs},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/KS6GT7IF}},
  note         = {Machine review of arXiv:1908.04320}
}
abstract

We study tropically planar graphs, which are the graphs that appear in smooth tropical plane curves. We develop necessary conditions for graphs to be tropically planar, and compute the number of tropically planar graphs up to genus $7$. We provide non-trivial upper and lower bounds on the number of tropically planar graphs, and prove that asymptotically $0\%$ of connected trivalent planar graphs are tropically planar.

Figures

Figures reproduced from arXiv: 1908.04320 by the authors.

Figure 1
Figure 1. A smooth tropical plane curve in the center, with its subdivided Newton polygon on the left and [PITH_FULL_IMAGE:figures/full_fig_p001_1.png] view at source ↗
Figure 2
Figure 2. Three lattice polygons; the first is hyperelliptic, and the first two are maximal [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. The process of inducing a regular triangulation with a height function; and a nonregular triangu [PITH_FULL_IMAGE:figures/full_fig_p003_3.png] view at source ↗
Figures from the paper (27 more)
Figure 4
Figure 4. Figure 4: The connected trivalent graphs of genus 2 and 3 [PITH_FULL_IMAGE:figures/full_fig_p004_4.png]
Figure 5
Figure 5. Figure 5: Newton subdivisions, tropical curves, and skeletons [PITH_FULL_IMAGE:figures/full_fig_p005_5.png]
Figure 6
Figure 6. Figure 6: A troplanar graph of genus 4, with a non-troplanar minor of genus 3 [PITH_FULL_IMAGE:figures/full_fig_p006_6.png]
Figure 7
Figure 7. Figure 7: Three genus 5 graphs easily shown not to be troplanar [PITH_FULL_IMAGE:figures/full_fig_p006_7.png]
Figure 8
Figure 8. Figure 8: Seven genus 5 graphs that are not troplanar [PITH_FULL_IMAGE:figures/full_fig_p007_8.png]
Figure 9
Figure 9. Figure 9: The form of a TIE-fighter graph graph is a TIE-fighter graph. The planar embedding of G coming from T must have C as a bounded face with the bridges incident to v1 and v2 exterior to it: otherwise the embedding would be crowded. The face formed by C is thus dual to som…
Figure 10
Figure 10. Figure 10: Illustrations for the two cases of our proof [PITH_FULL_IMAGE:figures/full_fig_p008_10.png]
Figure 11
Figure 11. Figure 11: A graph that is a TIE-fighter if g ≥ 5 We now present two more results, both involving bridges, that relate the troplanarity of different graphs to one another. Proposition 3.6. Let G be a troplanar graph with a bridge e. Then the connected components of G \ {e}, afte…
Figure 12
Figure 12. Figure 12: Splitting a polygon into smaller polygons, and splitting a troplanar graph into smaller troplanar [PITH_FULL_IMAGE:figures/full_fig_p009_12.png]
Figure 13
Figure 13. Figure 13: Bridge reduction Proposition 3.7. Let G be a troplanar graph with a bridge b. Then the bridge reduction of G with respect to b is troplanar. Proof. For a height function ω, write T (ω) for the triangulation induced by ω. Let ∆ be a lattice polygon and ω a height funct…
Figure 14
Figure 14. Figure 14: A bistellar flip The split between a and b subdivides ∆ into two polygons, ∆1 and ∆2. Let ω1 = ω|∆1 and ω2 = ω|∆2 be the restrictions of ω to these polygons. Note that the triangulation T (ωi) is simply the triangulation T (ω) restricted to ∆i . ∆ ∆1 ∆2 c d c d ∆ c d …
Figure 15
Figure 15. Figure 15: The starting triangulation of ∆; the restricted triangulations of ∆ [PITH_FULL_IMAGE:figures/full_fig_p010_15.png]
Figure 16
Figure 16. Figure 16: The maximal nonhyperelliptic polygons of genus 6 and genus 7 [PITH_FULL_IMAGE:figures/full_fig_p011_16.png]
Figure 17
Figure 17. Figure 17: All non-troplanar genus 6 graphs which are not ruled by any known criterion [PITH_FULL_IMAGE:figures/full_fig_p012_17.png]
Figure 18
Figure 18. Figure 18: The graph H used in the proof of Theorem 4.2 It follows from the above argument that most connected trivalent planar graphs (simple or otherwise) are sprawling. Choosing different H’s can similarly show that most such graphs are crowded, and that most such graphs are …
Figure 19
Figure 19. Figure 19: Certainly it is a subdivison: any choice of non-crossing edges between the top and bottom rows [PITH_FULL_IMAGE:figures/full_fig_p014_19.png]
Figure 19
Figure 19. Figure 19: A triangulation T of a lattice width 3 polygon ∆; the subset T 0 ; and the resulting triangulation T 0 ∪ ∂ [PITH_FULL_IMAGE:figures/full_fig_p015_19.png]
Figure 20
Figure 20. Figure 20: The relative positions of p, q, r, and s. The polygon Σ is solid, and necessary edges in T are dashed. Ignoring for a moment whether (0, 0) connects to (0, 1) and whether (a, 0) connects to (b, 1) in T , this means that the number of ways the upper and lower lattice p…
Figure 21
Figure 21. Figure 21: The three chains of genus three, and triangulations of a hyperelliptic polygon giving rise to them [PITH_FULL_IMAGE:figures/full_fig_p019_21.png]
Figure 22
Figure 22. Figure 22: The parallelogram P || g We will refer to the triangulated copies of P || 2 (respectively of P || 4 and P || 6 ) as tiles of genus 2 (respectively tiles of genus 4 and tiles of genus 6). To start out, we will use only one tile of genus 2, namely the one illustrated in…
Figure 23
Figure 23. Figure 23: The tile of genus 2, a dual tropical curve, and the troplanar graph with two marked points [PITH_FULL_IMAGE:figures/full_fig_p021_23.png]
Figure 24
Figure 24. Figure 24: The first eight tiles of genus 4 An example of a tiling of P || 14 is presented in the top left [PITH_FULL_IMAGE:figures/full_fig_p021_24.png]
Figure 25
Figure 25. Figure 25: A tiling of P || 14, and the corresponding troplanar graph; followed by a tiling of Qodd 7 (0, 3) (2, 0) (n + 3, 0) (n + 2, 3) (0, 3) (2, 0) (n + 3, 0) (n + 2, 3) (0, 1) [PITH_FULL_IMAGE:figures/full_fig_p022_25.png]
Figure 26
Figure 26. Figure 26: The polygons Qodd n and Qeven n , partially triangulated By construction, both G and H are of the form in Proposition 5.1, where G1, · · · , Gk are the graphs arising from the tiles T1, · · · , Tk, and H1, · · · , Hm are the graphs arising from the tiles T 0 1 , · · ·…
Figure 27
Figure 27. Figure 27: The tiles of genus 2 and 4 giving graphs with bridges [PITH_FULL_IMAGE:figures/full_fig_p023_27.png]
Figure 28
Figure 28. Figure 28: The tiles of genus 6 giving graphs without bridges [PITH_FULL_IMAGE:figures/full_fig_p027_28.png]
Figure 29
Figure 29. Figure 29: The tiles of genus 6 giving graphs with bridges [PITH_FULL_IMAGE:figures/full_fig_p028_29.png]

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