Perfect matchings with crossings
For sets of n points, n even, 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 Cn/2 different plane perfect matchings, where Cn/2 is the n/2-th Catalan number. Generalizing this result we are interested...
| Autores: | , , , , , , , |
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| Tipo de recurso: | artículo |
| Fecha de publicación: | 2023 |
| País: | España |
| Institución: | Universitat Politècnica de Catalunya (UPC) |
| Repositorio: | UPCommons. Portal del coneixement obert de la UPC |
| Idioma: | inglés |
| OAI Identifier: | oai:upcommons.upc.edu:2117/392579 |
| Acceso en línea: | https://hdl.handle.net/2117/392579 https://dx.doi.org/10.1007/s00453-023-01147-7 |
| Access Level: | acceso abierto |
| Palabra clave: | Combinatorial geometry Perfect matchings Crossings Geometric graphs Order types Geometria combinatòria Àrees temàtiques de la UPC::Matemàtiques i estadística::Geometria |
| Sumario: | For sets of n points, n even, 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 Cn/2 different plane perfect matchings, where Cn/2 is the n/2-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=164n2-3532nn--v+122564n , any set with 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 at most 572n2-n4 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. |
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