Perfect Matchings with Crossings

Oswin Aichholzer, Ruy Fabila-Monroy, Philipp Kindermann, Irene Parada, Rosna Paul*, Daniel Perz, Patrick Schnider, Birgit Vogtenhuber

*Corresponding author for this work

Research output: Chapter in Book/Report/Conference proceedingConference paperpeer-review

Abstract

For sets of n= 2 m points in general position in the plane we consider straight-line drawings of perfect matchings on them. It is well known that such sets admit at least C m different plane perfect matchings, where C m is the m-th Catalan number. Generalizing this result we are interested in the number of drawings of perfect matchings which have k crossings. We show the following results. (1) For every k≤164n2-O(nn), any set of n points, n sufficiently large, admits a perfect matching with exactly k crossings. (2) There exist sets of n points where every perfect matching has fewer than 572n2 crossings. (3) The number of perfect matchings with at most k crossings is superexponential in n if k is superlinear in n. (4) Point sets in convex position minimize the number of perfect matchings with at most k crossings for k= 0, 1, 2, and maximize the number of perfect matchings with (n/22) crossings and with (n/22)-1 crossings.

Original languageEnglish
Title of host publicationCombinatorial Algorithms
Subtitle of host publication33rd International Workshop, IWOCA 2022, Trier, Germany, June 7–9, 2022, Proceedings
EditorsCristina Bazgan, Henning Fernau
Place of PublicationCham
PublisherSpringer
Pages46-59
Number of pages14
ISBN (Electronic)978-3-031-06678-8
ISBN (Print)978-3-031-06677-1
DOIs
Publication statusPublished - 2022
Event33rd International Workshop on Combinatorial Algorithms: IWOCA 2022 - Trier, Germany
Duration: 7 Jun 20229 Jun 2022

Publication series

NameLecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
Volume13270 LNCS
ISSN (Print)0302-9743
ISSN (Electronic)1611-3349

Conference

Conference33rd International Workshop on Combinatorial Algorithms
Country/TerritoryGermany
CityTrier
Period7/06/229/06/22

Keywords

  • Perfect matchings
  • crossings
  • Combinatorial geometry
  • Order types
  • Crossings

ASJC Scopus subject areas

  • Theoretical Computer Science
  • Computer Science(all)

Fields of Expertise

  • Information, Communication & Computing

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