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Methods for Performing Crosses in Setaria viridis, a New Model System for the Grasses

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Published on: October 1, 2013

Perfect Matchings with Crossings.

Oswin Aichholzer¹, Ruy Fabila-Monroy², Philipp Kindermann³

¹Institute of Software Technology, Graz University of Technology, Inffeldgasse 16b, 8010 Graz, Austria.

|March 14, 2024

Summary

This summary is machine-generated.

This study explores perfect matchings in geometric point sets, proving that for large n, perfect matchings with any number of crossings k exist. Convex point sets minimize and maximize crossing counts for specific k values.

Keywords:

Combinatorial geometry Crossings Geometric graphs Order types Perfect matchings

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Area of Science:

Computational Geometry
Combinatorics
Graph Theory

Background:

Perfect matchings are pairs of non-adjacent vertices in a graph.
Plane perfect matchings are drawings of perfect matchings without edge crossings.
Catalan numbers quantify the number of plane perfect matchings for n points in convex position.

Purpose of the Study:

To generalize the understanding of perfect matchings beyond plane configurations.
To investigate the number of perfect matchings with a specific number of edge crossings (k).
To determine how point set configurations influence the distribution of crossing numbers in perfect matchings.

Main Methods:

Analysis of straight-line drawings of perfect matchings on sets of n points in general position.
Combinatorial arguments to establish existence and bounds on the number of crossings.
Comparison of crossing number distributions for point sets in general versus convex positions.

Main Results:

For sufficiently large n, any set of points in general position admits a perfect matching with exactly k crossings for any k.
Existence of point sets where all perfect matchings have at most O(n^2) crossings.
The number of perfect matchings with at most k crossings grows superexponentially with n when k is superlinear in n.
Point sets in convex position minimize the count of perfect matchings with at most k crossings and maximize those with k crossings.

Conclusions:

The number of crossings in perfect matchings is highly dependent on the configuration of the point set.
General position point sets offer greater flexibility in achieving specific crossing numbers.
Convex position point sets represent extremal cases for minimizing or maximizing matchings with certain crossing counts.