On the Richter-Thomassen conjecture about pairwise intersecting closed curves

Janos Pach, Natan Rubin, G. Tardos

Research output: Chapter in Book/Report/Conference proceedingConference contribution

6 Citations (Scopus)

Abstract

A long standing conjecture of Richter and Thomassen states that the total number of intersection points between any n simple closed Jordan curves in the plane, so that any two of them intersect and no three curves pass through the same point, is at least (1-O (1))n2. We confirm the above conjecture in several important cases, including the case (1) when all curves are convex, and (2) when the family of curves can be partitioned into two equal classes such that each curve from the first class is touching every curve from the second class. (Two curves are said to be touching if they have precisely one point in common, at which they do not properly cross.) An important ingredient of our proofs is the following statement: Let S be a family of the graphs of n continuous real functions defined on ℝ, no three of which pass through the same point. If there are nt pairs of touching curves in S, then the number of crossing points is ω (nt √logt/log log t).

Original languageEnglish
Title of host publicationProceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms
PublisherAssociation for Computing Machinery
Pages1506-1516
Number of pages11
Volume2015-January
EditionJanuary
Publication statusPublished - 2015
Event26th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2015 - San Diego, United States
Duration: Jan 4 2015Jan 6 2015

Other

Other26th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2015
CountryUnited States
CitySan Diego
Period1/4/151/6/15

Fingerprint

Closed curve
Pairwise
Curve
Jordan Curve
Intersect
Intersection
Graph in graph theory
Class

ASJC Scopus subject areas

  • Software
  • Mathematics(all)

Cite this

Pach, J., Rubin, N., & Tardos, G. (2015). On the Richter-Thomassen conjecture about pairwise intersecting closed curves. In Proceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms (January ed., Vol. 2015-January, pp. 1506-1516). Association for Computing Machinery.

On the Richter-Thomassen conjecture about pairwise intersecting closed curves. / Pach, Janos; Rubin, Natan; Tardos, G.

Proceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms. Vol. 2015-January January. ed. Association for Computing Machinery, 2015. p. 1506-1516.

Research output: Chapter in Book/Report/Conference proceedingConference contribution

Pach, J, Rubin, N & Tardos, G 2015, On the Richter-Thomassen conjecture about pairwise intersecting closed curves. in Proceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms. January edn, vol. 2015-January, Association for Computing Machinery, pp. 1506-1516, 26th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2015, San Diego, United States, 1/4/15.
Pach J, Rubin N, Tardos G. On the Richter-Thomassen conjecture about pairwise intersecting closed curves. In Proceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms. January ed. Vol. 2015-January. Association for Computing Machinery. 2015. p. 1506-1516
Pach, Janos ; Rubin, Natan ; Tardos, G. / On the Richter-Thomassen conjecture about pairwise intersecting closed curves. Proceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms. Vol. 2015-January January. ed. Association for Computing Machinery, 2015. pp. 1506-1516
@inproceedings{e2201ca3c0fe4f4c9ffe1cba1f217775,
title = "On the Richter-Thomassen conjecture about pairwise intersecting closed curves",
abstract = "A long standing conjecture of Richter and Thomassen states that the total number of intersection points between any n simple closed Jordan curves in the plane, so that any two of them intersect and no three curves pass through the same point, is at least (1-O (1))n2. We confirm the above conjecture in several important cases, including the case (1) when all curves are convex, and (2) when the family of curves can be partitioned into two equal classes such that each curve from the first class is touching every curve from the second class. (Two curves are said to be touching if they have precisely one point in common, at which they do not properly cross.) An important ingredient of our proofs is the following statement: Let S be a family of the graphs of n continuous real functions defined on ℝ, no three of which pass through the same point. If there are nt pairs of touching curves in S, then the number of crossing points is ω (nt √logt/log log t).",
author = "Janos Pach and Natan Rubin and G. Tardos",
year = "2015",
language = "English",
volume = "2015-January",
pages = "1506--1516",
booktitle = "Proceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms",
publisher = "Association for Computing Machinery",
edition = "January",

}

TY - GEN

T1 - On the Richter-Thomassen conjecture about pairwise intersecting closed curves

AU - Pach, Janos

AU - Rubin, Natan

AU - Tardos, G.

PY - 2015

Y1 - 2015

N2 - A long standing conjecture of Richter and Thomassen states that the total number of intersection points between any n simple closed Jordan curves in the plane, so that any two of them intersect and no three curves pass through the same point, is at least (1-O (1))n2. We confirm the above conjecture in several important cases, including the case (1) when all curves are convex, and (2) when the family of curves can be partitioned into two equal classes such that each curve from the first class is touching every curve from the second class. (Two curves are said to be touching if they have precisely one point in common, at which they do not properly cross.) An important ingredient of our proofs is the following statement: Let S be a family of the graphs of n continuous real functions defined on ℝ, no three of which pass through the same point. If there are nt pairs of touching curves in S, then the number of crossing points is ω (nt √logt/log log t).

AB - A long standing conjecture of Richter and Thomassen states that the total number of intersection points between any n simple closed Jordan curves in the plane, so that any two of them intersect and no three curves pass through the same point, is at least (1-O (1))n2. We confirm the above conjecture in several important cases, including the case (1) when all curves are convex, and (2) when the family of curves can be partitioned into two equal classes such that each curve from the first class is touching every curve from the second class. (Two curves are said to be touching if they have precisely one point in common, at which they do not properly cross.) An important ingredient of our proofs is the following statement: Let S be a family of the graphs of n continuous real functions defined on ℝ, no three of which pass through the same point. If there are nt pairs of touching curves in S, then the number of crossing points is ω (nt √logt/log log t).

UR - http://www.scopus.com/inward/record.url?scp=84938242774&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84938242774&partnerID=8YFLogxK

M3 - Conference contribution

AN - SCOPUS:84938242774

VL - 2015-January

SP - 1506

EP - 1516

BT - Proceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms

PB - Association for Computing Machinery

ER -