On flips in planar matchings

Kód výsledku v IS VaVaI

<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F00216208%3A11320%2F21%3A10419981" target="_blank" >RIV/00216208:11320/21:10419981 - isvavai.cz</a>

Výsledek na webu

<a href="https://verso.is.cuni.cz/pub/verso.fpl?fname=obd_publikace_handle&handle=aNpvfvxPqo" target="_blank" >https://verso.is.cuni.cz/pub/verso.fpl?fname=obd_publikace_handle&handle=aNpvfvxPqo</a>

DOI - Digital Object Identifier

<a href="http://dx.doi.org/10.1016/j.dam.2020.10.018" target="_blank" >10.1016/j.dam.2020.10.018</a>

Jazyk výsledku

angličtina

Název v původním jazyce

On flips in planar matchings

Popis výsledku v původním jazyce

In this paper we investigate the structure of flip graphs on non-crossing perfect matchings in the plane. Specifically, consider all non-crossing straight-line perfect matchings on a set of 2n points that are placed equidistantly on the unit circle. A flip operation on such a matching replaces two matching edges that span an empty quadrilateral with the other two edges of the quadrilateral, and the flip is called centered if the quadrilateral contains the center of the unit circle. The graph Gn has those matchings as vertices, and an edge between any two matchings that differ in a flip, and it is known to have many interesting properties. In this paper we focus on the spanning subgraph Hn of Gn obtained by taking all edges that correspond to centered flips, omitting edges that correspond to non-centered flips. We show that the graph Hn is connected for odd n, but has exponentially many small connected components for even n, which we characterize and count via Catalan and generalized Narayana numbers. For odd n, we also prove that the diameter of Hn is linear in n. Furthermore, we determine the minimum and maximum degree of Hn for all n, and characterize and count the corresponding vertices. Our results imply the non-existence of certain rainbow cycles in Gn, and they resolve several open questions and conjectures raised in a recent paper by Felsner, Kleist, Mütze, and Sering.

Název v anglickém jazyce

On flips in planar matchings

Popis výsledku anglicky

In this paper we investigate the structure of flip graphs on non-crossing perfect matchings in the plane. Specifically, consider all non-crossing straight-line perfect matchings on a set of 2n points that are placed equidistantly on the unit circle. A flip operation on such a matching replaces two matching edges that span an empty quadrilateral with the other two edges of the quadrilateral, and the flip is called centered if the quadrilateral contains the center of the unit circle. The graph Gn has those matchings as vertices, and an edge between any two matchings that differ in a flip, and it is known to have many interesting properties. In this paper we focus on the spanning subgraph Hn of Gn obtained by taking all edges that correspond to centered flips, omitting edges that correspond to non-centered flips. We show that the graph Hn is connected for odd n, but has exponentially many small connected components for even n, which we characterize and count via Catalan and generalized Narayana numbers. For odd n, we also prove that the diameter of Hn is linear in n. Furthermore, we determine the minimum and maximum degree of Hn for all n, and characterize and count the corresponding vertices. Our results imply the non-existence of certain rainbow cycles in Gn, and they resolve several open questions and conjectures raised in a recent paper by Felsner, Kleist, Mütze, and Sering.

Druh

J_imp - Článek v periodiku v databázi Web of Science

CEP obor

—

OECD FORD obor

10201 - Computer sciences, information science, bioinformathics (hardware development to be 2.2, social aspect to be 5.8)

Projekt

<a href="/cs/project/GA19-08554S" target="_blank" >GA19-08554S: Struktury a algoritmy ve velmi symetrických grafech</a><br>

Návaznosti

P - Projekt vyzkumu a vyvoje financovany z verejnych zdroju (s odkazem do CEP)

Rok uplatnění

2021

Kód důvěrnosti údajů

S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů

Název periodika

Discrete Applied Mathematics

ISSN

0166-218X

e-ISSN

—

Svazek periodika

neuveden

Číslo periodika v rámci svazku

289

Stát vydavatele periodika

NL - Nizozemsko

Počet stran výsledku

19

Strana od-do

427-445

Kód UT WoS článku

000596823800037

EID výsledku v databázi Scopus

2-s2.0-85093840376