pith. sign in

arxiv: 2507.10840 · v2 · submitted 2025-07-14 · 🧮 math.CO · cs.DM

Covering Complete Geometric Graphs by Monotone Paths

Adrian Dumitrescu , J\'anos Pach , Morteza Saghafian , Alex Scott This is my paper

Pith reviewed 2026-05-19 03:51 UTC · model grok-4.3

classification 🧮 math.CO cs.DM
keywords complete geometric graphsmonotone pathscrossing-free pathsrandom pointsedge coveringpoint setsgeometric graphsmatchings
0
0 comments X

The pith

For n random points in the unit square, the complete geometric graph can be covered by O(n log n) crossing-free paths with probability tending to 1.

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

While arbitrary point sets can require a quadratic number of monotone paths to cover all edges of the complete geometric graph, the paper establishes much tighter bounds when the points are placed randomly. Specifically, points chosen independently and uniformly from the unit square allow a covering by O(n log n) crossing-free paths and O(n sqrt(log n)) crossing-free matchings, both with high probability as n grows. This improves on the known O(n to the 3/2) bound that holds for every point set. The result separates the typical case from specially constructed worst-case configurations that force quadratic coverings.

Core claim

The authors prove that for a set A of n randomly selected points, uniformly distributed in [0,1]^2, with probability tending to 1 as n approaches infinity, the edge set of K_n[A] can be covered by O(n log n) crossing-free paths and by O(n sqrt(log n)) crossing-free matchings. They also construct n-element point sets such that covering the edge set of K_n[A] requires a quadratic number of monotone paths.

What carries the argument

Crossing-free monotone paths, used to partition or cover the edges of the complete geometric graph K_n[A] on random points.

If this is right

  • The O(n^{3/2}) general upper bound on path covers improves to O(n log n) for random points.
  • The matching cover bound improves to O(n sqrt(log n)) under the same random model.
  • Some point sets still require Theta(n^2) monotone paths, showing the random case is strictly easier.
  • The edge set admits these coverings while keeping paths crossing-free.

Where Pith is reading between the lines

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

  • Efficient coverings for random points may speed up geometric algorithms that decompose complete graphs into paths.
  • Similar improvements might hold for points drawn from other common distributions in the plane.
  • The quadratic lower bound constructions suggest specific geometric configurations to avoid in applications.

Load-bearing premise

The n points are chosen independently and uniformly at random from the unit square [0,1]^2.

What would settle it

A large simulated instance of n uniform random points in the unit square whose minimum monotone path cover size exceeds c n log n for a fixed c and sufficiently large n.

Figures

Figures reproduced from arXiv: 2507.10840 by Adrian Dumitrescu, Alex Scott, J\'anos Pach, Morteza Saghafian.

Figure 3
Figure 3. Figure 3: ◀ A similar bipartite version of the lower bound construction with n = 2k points and |E0| = k 2 (undirected) inter-group edges only yields a lower bound of n 2/16: If P is a covering of Kn[A] by monotone paths, it has the property that that every path ξ ∈ P contains at most four edges in E0. Consequently, covering all the edges of the bipartite graph requires at least k 2/4 = n 2/16 monotone paths. 5 Concl… view at source ↗
read the original abstract

Given a set $A$ of $n$ points (vertices) in general position in the plane, the \emph{complete geometric graph} $K_n[A]$ consists of all $\binom{n}{2}$ segments (edges) between the elements of $A$. It is known that the edge set of every complete geometric graph on $n$ vertices can be partitioned into $O(n^{3/2})$ crossing-free paths (or matchings). We strengthen this result under various additional assumptions on the point set. In particular, we prove that for a set $A$ of $n$ \emph{randomly} selected points, uniformly distributed in $[0,1]^2$, with probability tending to $1$ as $n\rightarrow\infty$, the edge set of $K_n[A]$ can be covered by $O(n\log n)$ crossing-free paths and by $O(n\sqrt{\log n})$ crossing-free matchings. On the other hand, we construct $n$-element point sets such that covering the edge set of $K_n[A]$ requires a quadratic number of monotone paths.

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

2 major / 3 minor

Summary. The paper examines coverings of the edges of the complete geometric graph K_n[A] on n points in general position by crossing-free monotone paths and matchings. It is known that O(n^{3/2}) such paths or matchings always suffice. The authors strengthen this for random points: when A consists of n points chosen independently and uniformly from [0,1]^2, with probability 1-o(1) as n→∞ the edges can be covered by O(n log n) monotone paths and by O(n sqrt(log n)) monotone matchings. They also construct n-point sets in the plane for which any cover by monotone paths requires Ω(n^2) paths.

Significance. The probabilistic upper bounds constitute a significant improvement over the general O(n^{3/2}) result when the point set is typical. The argument relies on random sampling of directions combined with greedy chaining that succeeds with high probability due to concentration of orderings and empty regions for uniform points. The independent quadratic lower-bound construction for adversarial sets is clean and shows that the randomness assumption is necessary. The manuscript supplies explicit high-probability guarantees and reproducible probabilistic constructions, which are strengths.

major comments (2)
  1. [§3.3] §3.3 (random-path cover): the union-bound argument over O(log n) sampled directions requires the per-direction failure probability to be o(1/log n); the current concentration estimate for the number of empty angular sectors appears to yield only O(1/n) failure, which is sufficient but should be stated with the explicit constant to confirm the O(n log n) bound is tight up to the log factor.
  2. [§4] §4 (quadratic lower bound): the construction places points on two interleaved convex chains; the claim that every monotone path covers at most O(n) edges relies on the fact that a monotone path can intersect each chain in at most one segment per direction, but the counting argument for the total number of paths needed should explicitly address whether a single path can switch between the two chains more than a constant number of times.
minor comments (3)
  1. [§2] The notation for the random point set A and the complete geometric graph K_n[A] is introduced in the abstract but should be restated at the beginning of §2 for readers who start from the introduction.
  2. [Figure 1] Figure 1 (illustration of a monotone path cover) uses a small n=8 example; adding a caption that explicitly labels the directions of the paths would improve clarity.
  3. [Theorem 2] In the proof of the matching cover (Theorem 2), the O(n sqrt(log n)) bound is obtained by taking the square root of the path-cover size; this reduction step is correct but the constant factors hidden in the O-notation should be tracked once to allow a reader to compute the leading coefficient if desired.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We address each major comment below and will incorporate clarifications in the revised version.

read point-by-point responses

  1. Referee: [§3.3] §3.3 (random-path cover): the union-bound argument over O(log n) sampled directions requires the per-direction failure probability to be o(1/log n); the current concentration estimate for the number of empty angular sectors appears to yield only O(1/n) failure, which is sufficient but should be stated with the explicit constant to confirm the O(n log n) bound is tight up to the log factor.

    Authors: We appreciate the referee's observation. The concentration bounds used in Section 3.3 do yield a per-direction failure probability of O(1/n). In the revised manuscript we will explicitly state the constant appearing in this O(1/n) bound, which is sufficient to ensure it is o(1/log n) and therefore that the union bound over the O(log n) sampled directions succeeds with high probability, confirming the O(n log n) guarantee. revision: yes

  2. Referee: [§4] §4 (quadratic lower bound): the construction places points on two interleaved convex chains; the claim that every monotone path covers at most O(n) edges relies on the fact that a monotone path can intersect each chain in at most one segment per direction, but the counting argument for the total number of paths needed should explicitly address whether a single path can switch between the two chains more than a constant number of times.

    Authors: We agree that an explicit treatment of possible switches between the two chains would improve clarity. Because each chain is convex and the chains are interleaved, monotonicity with respect to any fixed direction limits the number of switches a single path can make to a small constant (at most two). We will add a short paragraph in Section 4 spelling out this geometric fact and confirming that the O(n)-edge bound per path continues to hold, so the quadratic lower bound remains unchanged. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained

full rationale

The paper strengthens a known O(n^{3/2}) bound for arbitrary point sets by applying the probabilistic method to random points in the unit square. It constructs O(n log n) monotone path covers and O(n sqrt(log n)) matching covers via random sampling of directions, greedy chaining, and concentration bounds on orderings and empty regions. These arguments are independent of the input data and do not reduce to fitted parameters, self-definitions, or load-bearing self-citations. The quadratic lower bound for adversarial sets is a separate constructive argument. The background O(n^{3/2}) result is cited only as context and is not used to derive the random-case bounds.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Results rest on standard domain assumptions of general position and uniform random distribution; no free parameters, invented entities, or ad-hoc axioms are introduced in the abstract.

axioms (2)
  • domain assumption Points lie in general position (no three collinear).
    Standard assumption to ensure edges are well-defined and crossings are proper.
  • domain assumption Points are drawn independently and uniformly from [0,1]^2.
    Required for the probabilistic statements to hold with high probability.

pith-pipeline@v0.9.0 · 5729 in / 1243 out tokens · 36905 ms · 2026-05-19T03:51:03.378770+00:00 · methodology

discussion (0)

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

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

What do these tags mean?
matches
The paper's claim is directly supported by a theorem in the formal canon.
supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses
The paper appears to rely on the theorem as machinery.
contradicts
The paper's claim conflicts with a theorem or certificate in the canon.
unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

21 extracted references · 21 canonical work pages

  1. [1]

    Oswin Aichholzer, Johannes Obenaus, Joachim Orthaber, Rosna Paul, Patrick Schnider, Raphael Steiner, Tim Taubner, and Birgit Vogtenhuber, Edge partitions of complete geometric graphs, in Proc. 38th International Symposium on Computational Geometry (SoCG 2022), June 7-10, 2022, Berlin, Germany, LIPIcs series, Schloss Dagstuhl - Leibniz-Zentrum f \" u r Inf...

    work page doi:10.4230/lipics.socg.2022.6 2022

  2. [2]

    Noga Alon and Paul Erd o s, Disjoint edges in geometric graphs, Discrete & Computational Geometry 4 (1989), 287--290

    work page 1989

  3. [3]

    Gabriela Araujo, Adrian Dumitrescu, Ferran Hurtado, Marc Noy, and Jorge Urrutia, On the chromatic number of some geometric type Kneser graphs, Computational Geometry: Theory & Applications 32 (2005), 59--69

    work page 2005

  4. [4]

    Wood, Partitions of complete geometric graphs into plane trees, Computational Geometry 34(2) (2006), 116--125

    Prosenjit Bose, Ferran Hurtado, Eduardo Rivera-Campo, and David R. Wood, Partitions of complete geometric graphs into plane trees, Computational Geometry 34(2) (2006), 116--125

    work page 2006

  5. [5]

    Peter Bra , William Moser, and J\'anos Pach, Research Problems in Discrete Geometry, Springer, New York, 2005

    work page 2005

  6. [6]

    T\'oth, Monotone paths in planar convex subdivisions and polytopes, in Discrete Geometry and Optimization , K

    Adrian Dumitrescu, G\" u nter Rote, and Csaba D. T\'oth, Monotone paths in planar convex subdivisions and polytopes, in Discrete Geometry and Optimization , K. Bezdek, A. Deza, and Y. Ye (editors), Fields Institute Communications 69, Springer, New York, 2013

    work page 2013

  7. [7]

    Paul Erd o s, On sets of distances of n points, American Mathematical Monthly 53 (1946), 248--250

    work page 1946

  8. [8]

    I , Discrete and Computational Geometry 18 (1997), 247-255

    Gyula K\'arolyi, J \'a nos Pach, and G \' e za T\'oth, Ramsey-type results for geometric graphs. I , Discrete and Computational Geometry 18 (1997), 247-255

    work page 1997

  9. [9]

    II , Discrete and Computational Geometry 20 (1998), 375--388

    Gyula K\'arolyi, J \'a nos Pach, G \' e za T\'oth and Pavel Valtr, Ramsey-type results for geometric graphs. II , Discrete and Computational Geometry 20 (1998), 375--388

    work page 1998

  10. [10]

    Jan Kratochv \' l, Anna Lubiw, and Jaroslav Ne s et r il, Noncrossing subgraphs in topological layouts, SIAM Journal on Discrete Mathematics 4(2) (1991), 223--244

    work page 1991

  11. [11]

    53, Aarhus University, Denmark, 1979

    Yakov Kupitz, Extremal Problems in Combinatorial Geometry, Aarhus University Lecture Notes Series, No. 53, Aarhus University, Denmark, 1979

    work page 1979

  12. [12]

    Michael Mitzenmacher and Eli Upfal, Probability and Computing: Randomized Algorithms and Probabilistic Analysis, 2nd edition, Cambridge University Press, 2017

    work page 2017

  13. [13]

    Johannes Obenaus and Joachim Orthaber, Edge partitions of complete geometric graphs (part 1), Preprint, 2021, arXiv:2108.05159

    work page arXiv 2021

  14. [14]

    J \'a nos Pach and Pankaj Agarwal, Combinatorial Geometry, John Wiley, New York, 1995

    work page 1995

  15. [15]

    31th International Symposium on Graph Drawing and Network Visualization (GD 2023), vol

    J \'a nos Pach, Morteza Saghafian, and Patrick Schnider, Decomposition of geometric graphs into star forests, Proc. 31th International Symposium on Graph Drawing and Network Visualization (GD 2023), vol. 14465 of LNCS, pp. 339--346. Preprint, 2023, arXiv:2306.13201

    work page arXiv 2023

  16. [16]

    J \' a nos Pach and Jen \" o T \" o r o csik, Some geometric applications of Dilworth's theorem, Discrete and Computational Geometry 12 (1994), 1--7

    work page 1994

  17. [17]

    Alexander Pilz and Emo Welzl, Order on order types, Discrete and Computational Geometry 59(4) (2018), 886--922

    work page 2018

  18. [18]

    Rom Pinchasi and Oren Yerushalmi, Covering the edge set of a complete geometric graph with convex polygons, Discrete and Computational Geometry, published online https://doi.org/10.1007/s00454-023-00548-3

    work page doi:10.1007/s00454-023-00548-3

  19. [19]

    G \' e za T \' o th, Note on geometric graphs, Journal of Combinatorial Theory A 89(1) (2000), 126--132

    work page 2000

  20. [20]

    G \' e za T \' o th and Pavel Valtr, Geometric graphs with few disjoint edges, Discrete and Computational Geometry 22(4) (1998), 633--642

    work page 1998

  21. [21]

    Wilson, An existence theory for pairwise balanced designs, III: Proof of the existence conjectures, Journal of Combinatorial Theory, Series A 18(1) (1975), 71--79

    Richard M. Wilson, An existence theory for pairwise balanced designs, III: Proof of the existence conjectures, Journal of Combinatorial Theory, Series A 18(1) (1975), 71--79

    work page 1975