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On the edge-vertex ratio of maximal thrackles

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Pith's one-line read This paper constructs maximal thrackles with arbitrarily small edge-vertex ratio in the geometric case, and an infinite family without isolated vertices with ratio exactly 5/6 in the topological case.

desk verdict New extremal constructions for maximal thrackles, with the 5/6 family resting on a plausibly true but under-formalized belt construction. read the letter →

arxiv 1908.08857 v2 pith:RLDUAV4H submitted 2019-08-23 cs.DM cs.CGmath.CO

Oswin Aichholzer , Linda Kleist , Boris Klemz , Felix Schröder , Birgit Vogtenhuber This is my paper · ORCID
classification cs.DMcs.CGmath.CO MSC 05C1005C6205C35
keywords thracklemaximaledge-vertexratioconjecturegeometrictopologicalbeltconstructionsaturateddrawing
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

The paper asks how sparse a maximal thrackle can be: a thrackle is a drawing in which every pair of edges meets exactly once, and maximal means no edge between existing vertices can be added without breaking that condition. It shows that in the straight-line (geometric) setting the edge-vertex ratio can be made arbitrarily small by adding isolated vertices, and it can be pushed arbitrarily close to the natural lower bound of $1/2$ when isolated vertices are forbidden. For topological thrackles, isolated vertices again let the ratio tend to zero. The main theorem constructs an infinite family of maximal topological thrackles without isolated vertices whose edge-vertex ratio is exactly $5/6$. This matters because it shows that maximality does not force a thrackle to be dense, and it gives the lowest nondegenerate ratio currently exhibited toward the question of how low the ratio can go.

What carries the argument

The load-bearing mechanism is the belt construction: for each directed edge $e=uv$ of a cycle thrackle, place a copy of a four-edge, six-vertex local example with its vertices split between small disks around $u$ and $v$, and route every edge of the copy in a thin tunnel along $e$ so that it crosses all edges of the original drawing exactly once, while the two copies attached to consecutive edges also cross each other exactly once. The construction is engineered so that exactly four new edges and five new vertices are added per original edge, which fixes the edge-vertex ratio at $5/6$. The second mechanism is the maximality transfer encoded in Property 3: if a new edge can be added to the inflated drawing, it can be rerouted step by step until its endpoints lie in the underlying cycle, so maximality of the inflated drawing follows from maximality of the cycle.

What would settle it

Take the smallest case of the construction, $n=2$, for which $T_1$ is a 10-cycle, build an explicit drawing of $T_2$, and check every potential new edge between nonadjacent vertices: if any curve joins two such vertices while crossing every edge of $T_2$ exactly once, then $T_2$ is not maximal and Theorem 3 collapses. A systematic search over rotation systems of the 10-vertex drawing would settle the check.

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Extended reading notes

Core claim

The central discovery is the infinite family of Theorem 3: there exist maximal topological thrackles without isolated vertices with edge-vertex ratio exactly $5/6$. The proof begins with a star-shaped drawing of the odd cycle $C_{2n+1}$, duplicates every vertex and edge to obtain a maximal thrackle $T_1$ on the cycle $C_{4n+2}$, and then applies the belt construction to every edge of $T_1$. For each edge it places a copy of a fixed four-edge, six-vertex local example in a thin tunnel around that edge, interlacing the copy with the edge and its two neighbours so that every new edge crosses every other edge of the drawing exactly once. Each original edge survives and gains four new companion edges, while five new vertices are introduced, so the ratio $5/6$ follows by counting. The paper proves maximality of $T_2$ by a rerouting argument: any hypothetical new edge in $T_2$ can be locally rerouted to one whose endpoints belong to $T_1$, contradicting the known maximality of $T_1$.

Load-bearing premise

The construction assumes that all these small copies can be drawn simultaneously in thin tunnels around the edges so that every pair of edges crosses exactly once and no unintended intersections appear; the paper demonstrates the required interlacing in figures but does not give a formal proof that the simultaneous placement is always achievable.

Editorial extensions

If this is right

  • Maximality does not force a thrackle to have as many edges as vertices; maximal topological thrackles without isolated vertices can have ratio $5/6$, and geometric ones can approach $1/2$.
  • The lower bound $1/2$ from the handshaking lemma is asymptotically tight for maximal geometric thrackles without isolated vertices.
  • Adding isolated vertices is enough to drive the ratio to zero in both geometric and topological settings, so any positive lower bound for maximal thrackles must exclude isolated vertices.
  • The belt construction preserves maximality while inflating the edge count, giving a local operation that builds larger maximal thrackles from smaller ones.
  • Iterating the same construction on the original edges is proposed in the paper as a route toward ratios approaching $4/5$, which would leave the gap between $1/2$ and $5/6$ open.

Reading between the lines

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

  • The paper states the iterated belt construction as ongoing work; this reader's extrapolation is that the exact counting would give $4/5$ per additional round, but the difficulty is the maximality transfer, not the ratio.
  • The rerouting strategy suggests a reusable design principle: if a small, non-extendable local drawing is placed in a tunnel around each edge of a maximal thrackle, and every hypothetical new edge can be pulled back into the underlying cycle, then the inflated drawing inherits maximality. Testing this principle on other local modules could produce ratios below $5/6$.
  • A natural next experiment is to replace the four-edge, six-vertex local module by other non-extendable drawings with fewer edges per vertex and check whether the belt construction still closes; the paper does not explore this.
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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

3 major / 6 minor

Summary. The paper studies maximal thrackles, drawings of graphs in which every pair of edges intersects exactly once (at a common vertex or at a proper crossing), and investigates the possible values of the edge-vertex ratio ε(T)=|E|/|V|. It proves three existence results: (1) geometric maximal thrackles can have arbitrarily small ε, and, if isolated vertices are forbidden, can have ε arbitrarily close to the handshaking bound 1/2; (2) topological maximal thrackles with isolated vertices can have arbitrarily small ε; and (3) there is an infinite family of maximal topological thrackles without isolated vertices with ε exactly 5/6. The main construction in Theorem 3 starts from a star-shaped drawing of an odd cycle, duplicates vertices and edges to obtain a maximal thrackled cycle T1, and then applies a 'Kynčl belt construction' that attaches a copy of Kynčl's four-edge maximal thrackle to each edge of T1. The resulting graph T2 is shown to be maximal by a sequence of rerouting lemmas and structural properties (Lemmas 2–7, Properties 1–3).

Significance. If Theorem 3 is accepted, it is a valuable extremal result: apart from the trivial K1,1, it gives the first infinite family of maximal thrackles without isolated vertices whose edge-vertex ratio is strictly below 1, and it does so through a flexible gadget construction that may be adaptable to other saturation questions. The paper also contains a self-contained proof that Kynčl's example is maximal and a detailed proof that the duplicated cycle T1 is maximal, along with an independent verification of the specific case where Conway's conjecture for n≤12 is invoked. The constructions are explicit and the intermediate lemmas are clearly stated. The main weakness is that the Kynčl belt construction is described informally, with reference to figures, rather than by a formal existence proof, and the later lemmas inherit this informality. The paper does not include machine-checked proofs, but the case analysis is extensive and appears internally coherent.

major comments (3)
  1. [Section 4, paragraph after Figure 17] The Kynčl belt construction is not proved to exist. The text states that the copies K_e are drawn in thin tunnels around each edge e and that 'This ensures three facts', but no argument is given that a simultaneous drawing with the required intersection pattern is realizable. In particular, the claims that each edge of K_e intersects each edge of K_f and K_g precisely once, and that each edge of K_e intersects each edge of every remaining Kynčl copy exactly once, are global assertions about the interaction of 4|E(T1)| new curves. The existence of pairwise disjoint vertex vicinities and of tunnels with the required crossing behavior needs a proof or at least a constructive ordering, such as an ε-tunnel argument. As written, Theorem 3 and all subsequent lemmas rest on an unverified geometric hypothesis.
  2. [Section 4, Lemmas 2–7 and Properties 1–3] The proofs of these results depend on the precise local layout of the Kynčl copies inside the vertex vicinities, but this layout is described only by figures and informal phrases such as 'as illustrated in Figure 17' and 'the red-shaded region in Figure 19'. For example, Lemma 3 uses regions R, L, and G without textual definitions, and Property 3 refers to the triangular region T_u that is only shown in Figure 11. Since these lemmas establish maximality of T2, the authors should provide a combinatorial description of the local drawing in each vicinity, including the cyclic order of edges around each vertex and the sectors through which edges leave the disk, so that the case analyses can be checked independently of the figures.
  3. [Section 4, Proposition 2] The maximality proof of T1, though extensive, is not fully formal. In Case 1, the region R and the face C are not precisely defined, and the claim that the new edge crosses the boundary of R an even number of times 'since it contains C' is stated without proof. Similar issues appear in Cases 3 and 4 with the definitions of lower, middle, and upper parts of edges. Because Proposition 2 is used in the concluding step of Theorem 3 via Property 3, the argument should be made fully rigorous, for example by defining the relevant regions explicitly and justifying the parity or crossing claims.
minor comments (6)
  1. [Introduction] In the discussion of k-simple graphs, 'th k-simple property' should read 'the k-simple property'.
  2. [Section 5] The phrase 'maximal trackles' appears twice; it should be 'maximal thrackles'.
  3. [Lemma 3] The phrase 'we apply the usual modification for removing multiple edge crossings' is vague; please specify how the modification works and why it preserves the thrackle condition.
  4. [Property 3] In the sentence about replacing sections of s, 'close to the boundary of DU' should be 'close to the boundary of D_u'.
  5. [Theorem 2, direct proof in Case 3] The argument uses the fact that a thrackle cannot contain a 4-cycle without proof or citation; this is a standard consequence of Woodall's characterization of thrackled cycles and should be stated explicitly.
  6. [Section 2, proof of Theorem 1(b)] The sentence 'It is clear that by adding any number of segments in this way, we obtain a thrackle' is not fully justified; a short explanation of why the new segments intersect each other exactly once would be helpful.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: all central constructions and maximality proofs are direct, with no fitted inputs or author-imposed uniqueness assumptions.

full rationale

I walked the derivation chain of each theorem. The paper's central results are existence constructions, and every load-bearing claim is either proved directly or reduced to an independently proven proposition. The Kyncl example is introduced as external prior work, but the paper explicitly proves that it is a maximal thrackle in Proposition 1 rather than importing maximality. Theorem 3's key reduction is Property 3, which shows that if T1 is maximal then T2 is maximal; this is not circular because Proposition 2 independently proves maximality of T1, and Property 3 is a rerouting argument that establishes a rigorous implication. The use of the Pammer bound for n <= 12 in Theorem 2 is accompanied by an explicit independent direct proof, so it is not load-bearing self-citation. No fitted values are renamed as predictions, no quantity is defined in terms of the claimed output, and no author-imposed uniqueness theorem forces the constructions. The geometric feasibility of the belt construction is asserted with reference to figures rather than fully formalized, but that is a completeness or robustness concern, not circularity: the assertion is not equivalent to the theorem's conclusion by construction. The edge-vertex ratios are computed by explicit vertex and edge counts, not by assuming the desired ratio. Overall, the derivation chain is self-contained against external benchmarks and exhibits no circular step.

Assumptions & free parameters 0 free parameters · 3 assumptions · 0 invented entities

The paper is a pure construction paper. No free parameters are fitted to data. The main unproved inputs are the asserted existence of the 6-cycle thrackle drawing in Figure 3 and the geometric realizability of the belt construction, both supported primarily by figures. Standard planar topology is assumed.

assumptions (3)
  • domain assumption The drawing in Figure 3 is a thrackle of a 6-cycle with a central triangular face f0 and three adjacent quadrilateral faces f1, f2, f3.
    Theorem 2 relies on the existence and specific face structure of this drawing; the text asserts it via the figure without a formal construction.
  • domain assumption The Kynčl belt construction is geometrically realizable: each copy K_e can be drawn in a thin tunnel around its edge e so that every edge of K_e intersects every other edge of T2 exactly once and copies do not interfere.
    This is the load-bearing premise of Theorem 3, illustrated in Figures 17-25 but not proven formally in the text.
  • standard math Standard planar topology facts, including Euler's formula and face-counting arguments used in the case analyses.
    Used implicitly throughout the maximality proofs, especially in Theorem 2 and Proposition 2.

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Pith. "Pith review of On the edge-vertex ratio of maximal thrackles." pith.science (2026). https://pith.science/paper/RLDUAV4H

@misc{pith2026190808857,
  author       = {Pith},
  title        = {Pith review of: On the edge-vertex ratio of maximal thrackles},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/RLDUAV4H}},
  note         = {Machine review of arXiv:1908.08857}
}
read the original abstract

A drawing of a graph in the plane is a thrackle if every pair of edges intersects exactly once, either at a common vertex or at a proper crossing. Conway's conjecture states that a thrackle has at most as many edges as vertices. In this paper, we investigate the edge-vertex ratio of maximal thrackles, that is, thrackles in which no edge between already existing vertices can be inserted such that the resulting drawing remains a thrackle. For maximal geometric and topological thrackles, we show that the edge-vertex ratio can be arbitrarily small. When forbidding isolated vertices, the edge-vertex ratio of maximal geometric thrackles can be arbitrarily close to the natural lower bound of 1/2. For maximal topological thrackles without isolated vertices, we present an infinite family with an edge-vertex ratio of 5/6.

Figures

Figures reproduced from arXiv: 1908.08857 by the authors.

Figure 1
Figure 1. The butterfly T (thick, dark edges). Important segments between nonadjacent vertices are indicated in light gray. The thrackle Ta is obtained by adding multiple isolated vertices in the region R. The lower endpoint of each edge belongs to the set {b1, b2, b3} of bottom vertices and the upper endpoint belongs to the set {t1, t2, . . . , t7} of top vertices. The endpoints of two independent edges b1t2 and b2t1 are the… view at source ↗
Figure 2
Figure 2. The thrackle Tb is obtained by adding several segments uivi . along the central edge. All upper endpoints ui are placed on the line through t1 and t2, and all lower endpoints vi are placed on the line through b1 and b2. For each index i, the slope s(uivi) is negative. Moreover, we have s(uivi) < s(ujvj ) for i < j. Suppose that the first i−1 segments have already been created for some i ≥ 1. Then we choose the slope… view at source ↗
Figure 3
Figure 3. The thrackle T. f0 f1 u a b c d f e C1 f2 f3 [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figures from the paper (19 more)
Figure 5
Figure 5. Figure 5: Case 2 in Theorem 2. f0 f1 u a b c d f e C3 [PITH_FULL_IMAGE:figures/full_fig_p007_5.png]
Figure 7
Figure 7. Figure 7: {u, v} = {d, f}. f0 f1 a b c d f e Γ [PITH_FULL_IMAGE:figures/full_fig_p008_7.png]
Figure 9
Figure 9. Figure 9: Kynˇcl’s example K. Proposition 1. Kynˇcl’s example K is a maximal thrackle. Proof. We label the vertices of K as depicted in [PITH_FULL_IMAGE:figures/full_fig_p008_9.png]
Figure 10
Figure 10. Figure 10: Star-shaped thrackle T. external internal u Tu [PITH_FULL_IMAGE:figures/full_fig_p009_10.png]
Figure 12
Figure 12. Figure 12: Step 1: Duplicating the vertices and edges. The tunnel of uv is depicted grey-shaded. For convenience we slightly bend the edges of T before duplicating. In the right figure, the original cycle is indicated with dotted lines. then crosses the edge vw of T as well as u…
Figure 11
Figure 11. Figure 11: We next show that every pair of edges e,e 0 in T1 intersects and hence T1 is a thrackle. Lemma 1. T1 is a thrackle. Proof. Denote with eo and e 0 o the edges in T from which e and e 0 , respectively, originated. We distinguish the following cases: Case 1: eo = e 0 o .…
Figure 14
Figure 14. Figure 14: Illustration of Case 2 of Proposition 2. In Case 2, the vertices so and to share an edge in T, that is, {s, t} = {vi , wi} for some i ∈ {1, 2} and some directed edge vw of T. Let j fulfill {i, j} = {1, 2}. Let u and x be the vertices preceeding v and succeeding w, res…
Figure 15
Figure 15. Figure 15: Since the edges incident to s = v1 and t cannot be intersected again, t s v1 v2 a1 a2 u1 u2 w1 w2 R1 R2 R3 [PITH_FULL_IMAGE:figures/full_fig_p011_15.png]
Figure 16
Figure 16. Figure 16: Illustration of Case 4 of Proposition 2. to t, e crosses the boundary of R an even number of times. Since e may cross the boundary of R only via the section of a; it does not intersect R. Hence, e intersects a in its upper part. The just considered edge a is incident …
Figure 17
Figure 17. Figure 17: Kynˇcl belt construction, the original edges (thick) are preserved. Precisely, the construction works as follows: for each vertex v of T1 there exists a small disk Dv containing v such that the intersection of Dv with T1 is a simple curve consisting of parts of the tw…
Figure 18
Figure 18. Figure 18: Region s is trapped in after leaving Re on the side of Du It remains to consider the case that u is a second copy. In this case, the edge e is internal. Consider the region R0 u bounded by parts of f, parts of the edges incident to r, and part of any edge h non-incide…
Figure 11
Figure 11. Figure 11: All edge bundles other than Bf , Be, and Bg are crossed fully in Re. By assumption, the edge xgzg of Bg is already crossed in Dv. It follows that Bj = Be. However, since e is internal, the bundle Be is not incident to the outer face, which [PITH_FULL_IMAGE:figures/fu…
Figure 19
Figure 19. Figure 19: Regions to enter Re; rerouting. s f g u, be v, bg ce xe ye ze ae xg cg yf zf af Du Dv R L [PITH_FULL_IMAGE:figures/full_fig_p016_19.png]
Figure 21
Figure 21. Figure 21: The two possibilities for s to start at ce [PITH_FULL_IMAGE:figures/full_fig_p017_21.png]
Figure 21
Figure 21. Figure 21: Again, we obtain a contradiction and so s cannot be incident to ce. Case 2: If s starts at xe, then it is impossible for s 0 to be between xeye and xeze in Re, since otherwise it is forced to intersect L. It follows that s intersects aece in Dv. However, this is only …
Figure 22
Figure 22. Figure 22: Proof. We refer to af bf and xf zf as the outer edges, and to af cf and xf yf as the inner edges. Note that within Du, the outer edges bound a region R enclosing (parts of) the inner edges; see [PITH_FULL_IMAGE:figures/full_fig_p018_22.png]
Figure 23
Figure 23. Figure 23: Illustration of Lemma 5 and Lemma 6. Lemma 6. Let e = uv be an edge of T1. Then s intersects either both e and aebe or none of them inside Du. For an illustration, consider [PITH_FULL_IMAGE:figures/full_fig_p019_23.png]
Figure 24
Figure 24. Figure 24: Illustration of Property 3. After this replacement, the new edge s 0 intersects the same set of edges as s. Therefore, T2 + s 0 is a thrackle. Moreover, the vertex U of s is replaced by the vertex u of s 0 where u is in T1. If V 6= v, we apply the same rerouting for t…
Figure 25
Figure 25. Figure 25: Applying the Kynˇcl belt construction multiple times. By repeating the procedure k times, we obtain a trackle Tk with ε(Tk) = 2n + 1 + 4k 2n + 1 + 5k = 4 5 + 2n + 1 10n + 5 + 25k < 4 5 + c ⇔ k > (1 − 5c)(2n + 1) 25c . Showing that Tk is (potentially) maximal is more i…

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