Pith. Sign in

RECORD

Crossing lemma for the odd-crossing number

PaperPDF from arXiv

data files

Paper breakdownThe authors' claims, conditions and measurements, as JSON Signed receiptProof this review has not been altered
X Bluesky LinkedIn Reddit Hacker News
János Karl, Géza Tóth

Combinatorics · Submitted 25 Aug 2022

abstract compiled · 2026-09-23 · no language-model review

The authors' main claim If a graph can be drawn in the plane such that every edge is crossed by at most one other edge {\em an odd number of times}, then it is called 1-odd-planar and it has at most $5n-9$ edges.

how it is stated · asserted 1.00 read the record →

Comment (0) Read on arXiv ↗
Cite ▾
arXiv ID2208.12140 (version 1)
Pith IDS73K45D4, Pith's permanent record of this paper and its review
CitationG\'eza T\'oth, J\'anos Karl. "Crossing lemma for the odd-crossing number." arXiv:2208.12140 (2022). https://pith.science/paper/S73K45D4
Claim paper

The compiled record

The paper's own sentences, arranged. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

Open the compiled record

Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-09-23: 1 request, 4,880 input tokens.

What the abstract itself states, as probabilities: claims first 0.07 · states limitation 0.42 · human participants 0.01 · controlled comparison 0.03 · randomized assignment 0.01 · states quantitative result 0.98 · compares with prior results 0.86 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; .

A graph is $1$-planar, if it can be drawn in the plane such that there is at most one crossing on every edge. It is known, that $1$-planar graphs have at most $4n-8$ edges. We prove the following odd-even generalization. If a graph can be drawn in the plane such that every edge is crossed by at most one other edge {\em an odd number of times}, then it is called 1-odd-planar and it has at most $5n-9$ edges. As a consequence, we improve the constant in the Crossing Lemma for the odd-crossing number, if adjacent edges cross an even number of times. We also give upper bound for the number of edges of $k$-odd-planar graphs.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#sentencerolethe authors' own workstates the main conclusionstrength
s0 A graph is $1$-planar, if it can be drawn in the plane such that there is at most one crossing on every edge. background 1.00 0.07 0.02 asserted 0.99
s1 It is known, that $1$-planar graphs have at most $4n-8$ edges. background 1.00 0.04 0.03 asserted 1.00
s2 We prove the following odd-even generalization. contribution 0.98 0.56 0.03 asserted 0.69
s3 If a graph can be drawn in the plane such that every edge is crossed by at most one other edge {\em an odd number of times}, then it is called 1-odd-planar and it has at most $5n-9$ edges. result 0.96 0.87 0.71 asserted 1.00
s4 As a consequence, we improve the constant in the Crossing Lemma for the odd-crossing number, if adjacent edges cross an even number of times. interpretation 0.52 0.84 0.16 asserted 0.98
s5 We also give upper bound for the number of edges of $k$-odd-planar graphs. contribution 0.66 0.94 0.08 asserted 0.99

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 1.00.

What was examined, and what was not

sources read
  • arXiv listing abstract v1
not examined
  • full text
  • figures and tables
  • references and citing sentences
  • relations to other papers
levels run
  • P abstract
not run
  • S sections
  • T body sentences
  • C citations
  • R cross-paper relations

Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

The paper
Original abstract

A graph is $1$-planar, if it can be drawn in the plane such that there is at most one crossing on every edge. It is known, that $1$-planar graphs have at most $4n-8$ edges. We prove the following odd-even generalization. If a graph can be drawn in the plane such that every edge is crossed by at most one other edge {\em an odd number of times}, then it is called 1-odd-planar and it has at most $5n-9$ edges. As a consequence, we improve the constant in the Crossing Lemma for the odd-crossing number, if adjacent edges cross an even number of times. We also give upper bound for the number of edges of $k$-odd-planar graphs.

Human review

How Pith checks papers → · Full audit of this paper →

How the literature uses it

What this paper does with the arXiv papers it cites

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

Discussion

Know something about this paper? Leave a comment, or highlight any passage above to comment on it.